probability problem solving questions

Probability Questions with Solutions

Tutorial on finding the probability of an event. In what follows, S is the sample space of the experiment in question and E is the event of interest. n(S) is the number of elements in the sample space S and n(E) is the number of elements in the event E.

Questions and their Solutions

Answers to the above exercises, more references and links.

High Impact Tutoring Built By Math Experts

Personalized standards-aligned one-on-one math tutoring for schools and districts

Free ready-to-use math resources

Hundreds of free math resources created by experienced math teachers to save time, build engagement and accelerate growth

15 Probability Questions And Practice Problems for Middle and High School: Harder Exam Style Questions Included

Beki Christian

Probability questions and probability problems require students to work out how likely it is that something is to happen. Probabilities can be described using words or numbers. Probabilities range from 0 to 1 and can be written as fractions, decimals or percentages .

Here you’ll find a selection of probability questions of varying difficulty showing the variety you are likely to encounter in middle school and high school, including several harder exam style questions.

What are some real life examples of probability?

The more likely something is to happen, the higher its probability. We think about probabilities all the time.

For example, you may have seen that there is a 20% chance of rain on a certain day or thought about how likely you are to roll a 6 when playing a game, or to win in a raffle when you buy a ticket.

15 Probability Questions Worksheet

Get this 15 probability questions for your middle and high school students. Allows students tackle problems at their own pace and includes an answer key!

How to calculate probabilities

The probability of something happening is given by:

We can also use the following formula to help us calculate probabilities and solve problems:

• Probability of something not occuring = 1 – probability of if occurring P(not\;A) = 1 - P(A)
• For mutually exclusive events: Probability of event A OR event B occurring = Probability of event A + Probability of event B P(A\;or\;B) = P(A)+P(B)
• For independent events: Probability of event A AND event B occurring = Probability of event A times probability of event B P(A\;and\;B) = P(A) × P(B)

Probability question: A worked example

Question: What is the probability of getting heads three times in a row when flipping a coin?

When flipping a coin, there are two possible outcomes – heads or tails. Each of these options has the same probability of occurring during each flip. The probability of either heads or tails on a single coin flip is ½.

Since there are only two possible outcomes and they have the same probability of occurring, this is called a binomial distribution.

Let’s look at the possible outcomes if we flipped a coin three times.

Let H=heads and T=tails.

The possible outcomes are: HHH, THH, THT, HTT, HHT, HTH, TTH, TTT

Each of these outcomes has a probability of ⅛.

Therefore, the probability of flipping a coin three times in a row and having it land on heads all three times is ⅛.

Middle school probability questions

In middle school, probability questions introduce the idea of the probability scale and the fact that probabilities sum to one. We look at theoretical and experimental probability as well as learning about sample space diagrams and venn diagrams.

6th grade probability questions

1. Which number could be added to this spinner to make it more likely that the spinner will land on an odd number than a prime number?

Currently there are two odd numbers and two prime numbers so the chances of landing on an odd number or a prime number are the same. By adding 3, 5 or 11 you would be adding one prime number and one odd number so the chances would remain equal.

By adding 9 you would be adding an odd number but not a prime number. There would be three odd numbers and two prime numbers so the spinner would be more likely to land on an odd number than a prime number.

2. Ifan rolls a fair dice, with sides labeled A, B, C, D, E and F. What is the probability that the dice lands on a vowel?

A and E are vowels so there are 2 outcomes that are vowels out of 6 outcomes altogether.

Therefore the probability is   \frac{2}{6} which can be simplified to \frac{1}{3} .

7th grade probability questions

3. Max tested a coin to see whether it was fair. The table shows the results of his coin toss experiment:

Heads          Tails

    26                  41

What is the relative frequency of the coin landing on heads?

Max tossed the coin 67 times and it landed on heads 26 times.

\text{Relative frequency (experimental probability) } = \frac{\text{number of successful trials}}{\text{total number of trials}} = \frac{26}{67}

4. Grace rolled two dice. She then did something with the two numbers shown. Here is a sample space diagram showing all the possible outcomes:

What did Grace do with the two numbers shown on the dice?

Add them together

Subtract the number on dice 2 from the number on dice 1

Multiply them

Subtract the smaller number from the bigger number

For each pair of numbers, Grace subtracted the smaller number from the bigger number.

For example, if she rolled a 2 and a 5, she did 5 − 2 = 3.

8th grade probability questions

5. Alice has some red balls and some black balls in a bag. Altogether she has 25 balls. Alice picks one ball from the bag. The probability that Alice picks a red ball is x and the probability that Alice picks a black ball is 4x. Work out how many black balls are in the bag.

Since the probability of mutually exclusive events add to 1:  

\begin{aligned} x+4x&=1\\\\ 5x&=1\\\\ x&=\frac{1}{5} \end{aligned}

\frac{1}{5} of the balls are red and \frac{4}{5} of the balls are blue.

6. Arthur asked the students in his class whether they like math and whether they like science. He recorded his results in the venn diagram below.

How many students don’t like science?

We need to look at the numbers that are not in the ‘Like science’ circle. In this case it is 9 + 7 = 16.

High school probability questions

In high school, probability questions involve more problem solving to make predictions about the probability of an event. We also learn about probability tree diagrams, which can be used to represent multiple events, and conditional probability.

9th grade probability questions

7. A restaurant offers the following options:

Starter – soup or salad

Main – chicken, fish or vegetarian

Dessert – ice cream or cake

How many possible different combinations of starter, main and dessert are there?

The number of different combinations is 2 × 3 × 2 = 12.

8. There are 18 girls and 12 boys in a class. \frac{2}{9} of the girls and \frac{1}{4} of the boys walk to school. One of the students who walks to school is chosen at random. Find the probability that the student is a boy. 

First we need to work out how many students walk to school:

\frac{2}{9} \text{ of } 18 = 4

\frac{1}{4} \text{ of } 12 = 3

7 students walk to school. 4 are girls and 3 are boys. So the probability the student is a boy is \frac{3}{7} .

9. Rachel flips a biased coin. The probability that she gets two heads is 0.16. What is the probability that she gets two tails?

We have been given the probability of getting two heads. We need to calculate the probability of getting a head on each flip.

Let’s call the probability of getting a head p.

The probability p, of getting a head AND getting another head is 0.16.

Therefore to find p:

The probability of getting a head is 0.4 so the probability of getting a tail is 0.6.

The probability of getting two tails is 0.6 × 0.6 = 0.36 .

10th grade probability questions

10. I have a big tub of jelly beans. The probability of picking each different color of jelly bean is shown below:

If I were to pick 60 jelly beans from the tub, how many orange jelly beans would I expect to pick?

First we need to calculate the probability of picking an orange. Probabilities sum to 1 so 1 − (0.2 + 0.15 + 0.1 + 0.3) = 0.25.

The probability of picking an orange is 0.25.

The number of times I would expect to pick an orange jelly bean is 0.25 × 60 = 15 .

11. Dexter runs a game at a fair. To play the game, you must roll a dice and pick a card from a deck of cards.

To win the game you must roll an odd number and pick a picture card. The game can be represented by the tree diagram below.

Dexter charges players $1 to play and gives $3 to any winners. If 260 people play the game, how much profit would Dexter expect to make?

Completing the tree diagram:

Probability of winning is \frac{1}{2} \times \frac{4}{13} = \frac{4}{26}

If 260 play the game, Dexter would receive $260.

The expected number of winners would be \frac{4}{26} \times 260 = 40

Dexter would need to give away 40 × $3 = $120 .

Therefore Dexter’s profit would be $260 − $120 = $140.

12. A fair coin is tossed three times. Work out the probability of getting two heads and one tail.

There are three ways of getting two heads and one tail: HHT, HTH or THH.

The probability of each is \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} 

Therefore the total probability is \frac{1}{8} +\frac{1}{8} + \frac{1}{8} = \frac{3}{8}

11th/12th grade probability questions

13. 200 people were asked about which athletic event they thought was the most exciting to watch. The results are shown in the table below.

A person is chosen at random. Given that that person chose 100m, what is the probability that the person was female?

Since we know that the person chose 100m, we need to include the people in that column only.

In total 88 people chose 100m so the probability the person was female is \frac{32}{88}   .

14.   Sam asked 50 people whether they like vegetable pizza or pepperoni pizza.

37 people like vegetable pizza. 

25 people like both. 

3 people like neither.

Sam picked one of the 50 people at random. Given that the person he chose likes pepperoni pizza, find the probability that they don’t like vegetable pizza.

We need to draw a venn diagram to work this out.

We start by putting the 25 who like both in the middle section. The 37 people who like vegetable pizza includes the 25 who like both, so 12 more people must like vegetable pizza. 3 don’t like either. We have 50 – 12 – 25 – 3 = 10 people left so this is the number that must like only pepperoni.

There are 35 people altogether who like pepperoni pizza. Of these, 10 do not like vegetable pizza. The probability is   \frac{10}{35} .

15. There are 12 marbles in a bag. There are n red marbles and the rest are blue marbles. Nico takes 2 marbles from the bag. Write an expression involving n for the probability that Nico takes one red marble and one blue marble.

We need to think about this using a tree diagram. If there are 12 marbles altogether and n are red then 12-n are blue.

To get one red and one blue, Nico could choose red then blue or blue then red so the probability is:

Looking for more middle school and high school probability math questions?

• Ratio questions
• Algebra questions
• Trigonometry questions
• Venn diagram questions
• Long division questions
• Pythagorean theorem questions

Do you have students who need extra support in math? Give your students more opportunities to consolidate learning and practice skills through personalized math tutoring with their own dedicated online math tutor. Each student receives differentiated instruction designed to close their individual learning gaps, and scaffolded learning ensures every student learns at the right pace. Lessons are aligned with your state’s standards and assessments, plus you’ll receive regular reports every step of the way. Personalized one-on-one math tutoring programs are available for: – 2nd grade tutoring – 3rd grade tutoring – 4th grade tutoring – 5th grade tutoring – 6th grade tutoring – 7th grade tutoring – 8th grade tutoring Why not learn more about how it works ?

The content in this article was originally written by secondary teacher Beki Christian and has since been revised and adapted for US schools by elementary math teacher Katie Keeton.

Related articles

15 Venn Diagram Questions And Practice Problems (Middle & High School): Exam Style Questions Included

9 Algebra Questions And Practice Problems To Do With Your Middle Schoolers

15 Trigonometry Questions And Practice Problems To Do With High Schoolers

Ratio Questions And Practice Problems: Differentiated Practice Questions Included

Pythagoras Theorem Questions [FREE]

Downloadable Pythagoras theorem worksheet containing 15 multiple choice questions with a mix of worded problems and deeper problem solving questions.

Includes an answer key and follows variation theory with plenty of opportunities for students to work independently at their own level.

Privacy Overview

Probability Word Problems

In these lessons, we will learn how to solve a variety of probability problems.

Related Pages Probability Tree Diagrams Probability Without Replacement Theoretical vs. Experimental Probability More Lessons On Probability

Here we shall be looking into solving probability word problems involving:

• Probability and Sample Space
• Probability and Frequency Table
• Probability and Area
• Probability of Simple Events
• Probability and Permutations
• Probability and Combinations
• Probability of Independent Events

We will now look at some examples of probability problems.

Example: At a car park there are 100 vehicles, 60 of which are cars, 30 are vans and the remainder are lorries. If every vehicle is equally likely to leave, find the probability of: a) a van leaving first. b) a lorry leaving first. c) a car leaving second if either a lorry or van had left first.

Solution: a) Let S be the sample space and A be the event of a van leaving first. n(S) = 100 n(A) = 30

c) If either a lorry or van had left first, then there would be 99 vehicles remaining, 60 of which are cars. Let T be the sample space and C be the event of a car leaving. n(T) = 99 n(C) = 60

Example: A survey was taken on 30 classes at a school to find the total number of left-handed students in each class. The table below shows the results:

A class was selected at random. a) Find the probability that the class has 2 left-handed students. b) What is the probability that the class has at least 3 left-handed students? c) Given that the total number of students in the 30 classes is 960, find the probability that a student randomly chosen from these 30 classes is left-handed.

a) Let S be the sample space and A be the event of a class having 2 left-handed students. n(S) = 30 n(A) = 5

b) Let B be the event of a class having at least 3 left-handed students. n(B) = 12 + 8 + 2 = 22

c) First find the total number of left-handed students:

Total no. of left-handed students = 2 + 10 + 36 + 32 + 10 = 90

Here, the sample space is the total number of students in the 30 classes, which was given as 960.

Let T be the sample space and C be the event that a student is left-handed. n(T) = 960 n(C) = 90

Probability And Area

Example: ABCD is a square. M is the midpoint of BC and N is the midpoint of CD. A point is selected at random in the square. Calculate the probability that it lies in the triangle MCN.

Area of square = 2x × 2x = 4x 2

This video shows some examples of probability based on area.

Probability Of Simple Events

The following video shows some examples of probability problems. A few examples of calculating the probability of simple events.

• What is the probability of the next person you meeting having a phone number that ends in 5?
• What is the probability of getting all heads if you flip 3 coins?
• What is the probability that the person you meet next has a birthday in February? (Non-leap year)

This video introduces probability and gives many examples to determine the probability of basic events.

A bag contains 8 marbles numbered 1 to 8 a. What is the probability of selecting a 2 from the bag? b. What is the probability of selecting an odd number? c. What is the probability of selecting a number greater than 6?

Using a standard deck of cards, determine each probability. a. P(face card) b. P(5) c. P(non face card)

Using Permutations To Solve Probability Problems

This video shows how to evaluate factorials, how to use permutations to solve probability problems, and how to determine the number of permutations with indistinguishable items.

A permutation is an arrangement or ordering. For a permutation, the order matters.

• If a class has 28 students, how many different arrangements can 5 students give a presentation to the class?
• How many ways can the letters of the word PHEONIX be arranged?
• How many ways can you order 3 blue marbles, 4 red marbles and 5 green marbles? Marbles of the same color look identical.

Using Combinations To Solve Probability Problems

This video shows how to evaluate combinations and how to use combinations to solve probability problems.

A combination is a grouping or subset of items. For a combination, the order does not matter.

• The soccer team has 20 players. There are always 11 players on the field. How many different groups of players can be in the field at the same time?
• A student needs 8 more classes to complete her degree. If she has met the prerequisites for all the courses, how many ways can she take 4 class next semester?
• There are 4 men and 5 women in a small office. The customer wants a site visit from a group of 2 men and 2 women. How many different groups van be formed from the office?

How To Find The Probability Of Different Events?

This video explains how to determine the probability of different events. This can be found that can be found using combinations and basic probability.

• The probability of drawing 2 cards that are both face cards.
• The probability of drawing 2 cards that are both aces.
• The probability of drawing 4 cards all from the same suite.

A group of 10 students made up of 6 females and 4 males form a committee of 4. What is the probability the committee is all male? What is the probability that the committee is all female? What is the probability the committee is made up of 2 females and 2 males?

How To Find The Probability Of Multiple Independent Events?

This video explains the counting principle and how to determine the number of ways multiple independent events can occur.

• How many ways can students answer a 3-question true of false quiz?
• How many passwords using 6 digits where the first digit must be letters and the last four digits must be numbers?
• A restaurant offers a dinner special in which you get to pick 1 item from 4 different categories. How many different meals are possible?
• A door lock on a classroom requires entry of 4 digits. All digits must be numbers, but the digits can not be repeated. How many unique codes are possible?

How To Find The Probability Of A Union Of Two Events?

This video shows how to determine the probability of a union of two events.

• If you roll 2 dice at the same time, what is the probability the sum is 6 or a pair of odd numbers?
• What is the probability of selecting 1 card that is red or a face card?

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

• Math Article

Probability Questions

The probability questions , with answers, are provided here for students to make them understand the concept in an easy way. The chapter Probability has been included in Class 9, 10, 11 and 12. Therefore, it is a very important chapter. The questions here will be provided, as per NCERT guidelines. Get Probability For Class 10 at BYJU’S.

The application of probability can be seen in Maths as well as in day to day life. It is necessary to learn the basics of this concept. The questions here will cover the basics as well as the hard level problems for all levels of students. Thus, students will be confident in solving problems based on it. Also, solving these probability problems will help them to participate in competitive exams, going further.

Definition: Probability is nothing but the possibility of an event occurring. For example, when a test is conducted, then the student can either get a pass or fail. It is a state of probability.

Also read: Probability

The probability of happening of an event E is a number P(E) such that:

0 ≤ P(E) ≤ 1

Probability Formula: If an event E occurs, then the empirical probability of an event to happen is:

P(E) = Number of trials in which Event happened/Total number of trials

The theoretical probability of an event E, P(E), is defined as:

P(E) = (Number of outcomes favourable to E)/(Number of all possible outcomes of the experiment)

Impossible event: The probability of an occurrence/event impossible to happen is 0. Such an event is called an impossible event.

Sure event: The probability of an event that is sure to occur is 1. Such an event is known as a sure event or a certain event.

Probability Questions & Answers

1. Two coins are tossed 500 times, and we get:

Two heads: 105 times

One head: 275 times

No head: 120 times

Find the probability of each event to occur.

Solution: Let us say the events of getting two heads, one head and no head by E 1 , E 2 and E 3 , respectively.

P(E 1 ) = 105/500 = 0.21

P(E 2 ) = 275/500 = 0.55

P(E 3 ) = 120/500 = 0.24

The Sum of probabilities of all elementary events of a random experiment is 1.

P(E 1 )+P(E 2 )+P(E 3 ) = 0.21+0.55+0.24 = 1

2. A tyre manufacturing company kept a record of the distance covered before a tyre needed to be replaced. The table shows the results of 1000 cases.

If a tyre is bought from this company, what is the probability that :

(i) it has to be substituted before 4000 km is covered?

(ii) it will last more than 9000 km?

(iii) it has to be replaced after 4000 km and 14000 km is covered by it?

Solution: (i) Total number of trials = 1000.

The frequency of a tyre required to be replaced before covering 4000 km = 20

So, P(E 1 ) = 20/1000 = 0.02

(ii) The frequency that tyre will last more than 9000 km = 325 + 445 = 770

So, P(E 2 ) = 770/1000 = 0.77

(iii) The frequency that tyre requires replacement between 4000 km and 14000 km = 210 + 325 = 535.

So, P(E 3 ) = 535/1000 = 0.535

3. The percentage of marks obtained by a student in the monthly tests are given below:

Based on the above table, find the probability of students getting more than 70% marks in a test.

Solution: The total number of tests conducted is 5.

The number of tests when students obtained more than 70% marks = 3.

So, P(scoring more than 70% marks) = ⅗ = 0.6

4. One card is drawn from a deck of 52 cards, well-shuffled. Calculate the probability that the card will

(i) be an ace,

(ii) not be an ace.

Solution: Well-shuffling ensures equally likely outcomes.

(i) There are 4 aces in a deck.

Let E be the event the card drawn is ace.

The number of favourable outcomes to the event E = 4

The number of possible outcomes = 52

Therefore, P(E) = 4/52 = 1/13

(ii) Let F is the event of ‘card is not an ace’

The number of favourable outcomes to F = 52 – 4 = 48

Therefore, P(F) = 48/52 = 12/13

5. Two players, Sangeet and Rashmi, play a tennis match. The probability of Sangeet winning the match is 0.62. What is the probability that Rashmi will win the match?

Solution: Let S and R denote the events that Sangeeta wins the match and Reshma wins the match, respectively.

The probability of Sangeet to win = P(S) = 0.62

The probability of Rashmi to win = P(R) = 1 – P(S)

= 1 – 0.62 = 0.38

6. Two coins (a one rupee coin and a two rupee coin) are tossed once. Find a sample space.

Solution: Either Head(H) or Tail(T) can be the outcomes.

Heads on both coins = (H,H) = HH

Head on 1st coin and Tail on the 2nd coin = (H,T) = HT

Tail on 1st coin and Head on the 2nd coin = (T,H) = TH

Tail on both coins = (T,T) = TT

Therefore, the sample space is S = {HH, HT, TH, TT}

7. Consider the experiment in which a coin is tossed repeatedly until a head comes up. Describe the sample space.

Solution: In the random experiment where the head can appear on the 1st toss, or the 2nd toss, or the 3rd toss and so on till we get the head of the coin. Hence, the required sample space is :

S= {H, TH, TTH, TTTH, TTTTH,…}

8. Consider the experiment of rolling a die. Let A be the event ‘getting a prime number’, B be the event ‘getting an odd number’. Write the sets representing the events

(ii) A and B

(iii) A but not B

(iv) ‘not A’.

Solution: S = {1, 2, 3, 4, 5, 6}, A = {2, 3, 5} and B = {1, 3, 5}

(i) A or B = A ∪ B = {1, 2, 3, 5}

(ii) A and B = A ∩ B = {3,5}

(iii) A but not B = A – B = {2}

(iv) not A = A′ = {1,4,6}

9. A coin is tossed three times, consider the following events.

P: ‘No head appears’,

Q: ‘Exactly one head appears’ and

R: ‘At Least two heads appear’.

Check whether they form a set of mutually exclusive and exhaustive events.

Solution: The sample space of the experiment is:

S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} and

Q = {HTT, THT, TTH},

R = {HHT, HTH, THH, HHH}

P ∪ Q ∪ R = {TTT, HTT, THT, TTH, HHT, HTH, THH, HHH} = S

Therefore, P, Q and R are exhaustive events.

P ∩ R = φ and

Therefore, the events are mutually exclusive.

Hence, P, Q and R form a set of mutually exclusive and exhaustive events.

10. If P(A) = 7/13, P(B) = 9/13 and P(A∩B) = 4/13, evaluate P(A|B).

Solution: P(A|B) = P(A∩B)/P(B) = (4/13)/(9/13) = 4/9.

Video Lesson

Probability important topics.

Probability Important Questions

Related Links

• Important Questions Class 9 Maths Chapter 15 Probability
• Important Questions Class 10 Maths Chapter 15 Probability
• Important Questions Class 11 Maths Chapter 16 Probability
• Important Questions Class 12 Maths Chapter 13 Probability

Practice Questions

Solve the following probability questions.

• Write the sample space for rolling two dice.
• If two coins are tossed simultaneously, what is the probability of getting exactly two heads?
• From a well-shuffled deck of 52 cards, what is the probability of getting a king?
• In a bag, there are 5 red balls and 7 black balls. What is the probability of getting a black ball?
• If the probability of an event happening is 0.7, then what is the probability of an event that will not happen?

Stay tuned with BYJU’S – The Learning App and learn all Maths-related concepts easily by exploring more videos.

Leave a Comment Cancel reply

Your Mobile number and Email id will not be published. Required fields are marked *

Request OTP on Voice Call

Post My Comment

• Share Share

Register with BYJU'S & Download Free PDFs

Register with byju's & watch live videos.

Teach yourself statistics

How to Solve Probability Problems

You can solve many simple probability problems just by knowing two simple rules:

• The probability of any sample point can range from 0 to 1.
• The sum of probabilities of all sample points in a sample space is equal to 1.

The following sample problems show how to apply these rules to find (1) the probability of a sample point and (2) the probability of an event.

Probability of a Sample Point

The probability of a sample point is a measure of the likelihood that the sample point will occur.

Example 1 Suppose we conduct a simple statistical experiment . We flip a coin one time. The coin flip can have one of two equally-likely outcomes - heads or tails. Together, these outcomes represent the sample space of our experiment. Individually, each outcome represents a sample point in the sample space. What is the probability of each sample point?

Solution: The sum of probabilities of all the sample points must equal 1. And the probability of getting a head is equal to the probability of getting a tail. Therefore, the probability of each sample point (heads or tails) must be equal to 1/2.

Example 2 Let's repeat the experiment of Example 1, with a die instead of a coin. If we toss a fair die, what is the probability of each sample point?

Solution: For this experiment, the sample space consists of six sample points: {1, 2, 3, 4, 5, 6}. Each sample point has equal probability. And the sum of probabilities of all the sample points must equal 1. Therefore, the probability of each sample point must be equal to 1/6.

Probability of an Event

The probability of an event is a measure of the likelihood that the event will occur. By convention, statisticians have agreed on the following rules.

• The probability of any event can range from 0 to 1.
• The probability of event A is the sum of the probabilities of all the sample points in event A.
• The probability of event A is denoted by P(A).

Thus, if event A were very unlikely to occur, then P(A) would be close to 0. And if event A were very likely to occur, then P(A) would be close to 1.

Example 1 Suppose we draw a card from a deck of playing cards. What is the probability that we draw a spade?

Solution: The sample space of this experiment consists of 52 cards, and the probability of each sample point is 1/52. Since there are 13 spades in the deck, the probability of drawing a spade is

P(Spade) = (13)(1/52) = 1/4

Example 2 Suppose a coin is flipped 3 times. What is the probability of getting two tails and one head?

Solution: For this experiment, the sample space consists of 8 sample points.

S = {TTT, TTH, THT, THH, HTT, HTH, HHT, HHH}

Each sample point is equally likely to occur, so the probability of getting any particular sample point is 1/8. The event "getting two tails and one head" consists of the following subset of the sample space.

A = {TTH, THT, HTT}

The probability of Event A is the sum of the probabilities of the sample points in A. Therefore,

P(A) = 1/8 + 1/8 + 1/8 = 3/8

Probability Practice Questions

Click here for questions, click here for answers, gcse revision cards.

5-a-day Workbooks

Primary Study Cards

Privacy Policy

Terms and Conditions

Corbettmaths © 2012 – 2024

• Testimonial
• Web Stories

Learning Home

Not Now! Will rate later

• Probability: Solved Examples

• Probability: Concepts & Tricks
• Probability Practice Questions: Level 01
• Probability Practice Questions: Level 02

Most Popular Articles - PS

Time and Work Concepts

Time and Work Formula and Solved Problems

Time and Work Problems (Easy)

Time and Work Problems (Difficult)

Problems on Ages Practice Problems : Level 02

Chain Rule : Theory & Concepts

Chain Rule Solved Examples

Chain Rule Practice Problems: Level 01

Chain Rule Practice Problems : Level 02

Problems on Numbers System : Level 02

Download our app.

• Learn on-the-go
• Unlimited Prep Resources
• Better Learning Experience
• Personalized Guidance

Get More Out of Your Exam Preparation - Try Our App!

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

Unit 7: Probability

About this unit.

If you're curious about the mathematical ins and outs of probability, you've come to the right unit! Here, we'll take a deep dive into the many ways we can calculate the likelihood of different outcomes. From using simulations to the addition and multiplication rules, we'll build a solid foundation that will help us tackle statistical questions down the line.

Estimating probabilities using simulation

• Intro to theoretical probability (Opens a modal)
• Experimental versus theoretical probability simulation (Opens a modal)
• Theoretical and experimental probability: Coin flips and die rolls (Opens a modal)
• Random number list to run experiment (Opens a modal)
• Random numbers for experimental probability (Opens a modal)
• Interpret results of simulations Get 3 of 4 questions to level up!

Mutually exclusive events and unions of events

• Probability with Venn diagrams (Opens a modal)
• Addition rule for probability (Opens a modal)
• Addition rule for probability (basic) (Opens a modal)
• Two-way tables, Venn diagrams, and probability Get 3 of 4 questions to level up!

Conditional probability

• Conditional probability and independence (Opens a modal)
• Conditional probability with Bayes' Theorem (Opens a modal)
• Conditional probability using two-way tables (Opens a modal)
• Conditional probability tree diagram example (Opens a modal)
• Tree diagrams and conditional probability (Opens a modal)
• Calculate conditional probability Get 3 of 4 questions to level up!

Independent versus dependent events and the multiplication rule

• Compound probability of independent events (Opens a modal)
• Independent events example: test taking (Opens a modal)
• General multiplication rule example: independent events (Opens a modal)
• Dependent probability introduction (Opens a modal)
• General multiplication rule example: dependent events (Opens a modal)
• The general multiplication rule (Opens a modal)
• "At least one" probability with coin flipping (Opens a modal)
• Probabilities involving "at least one" success (Opens a modal)
• Probability with general multiplication rule Get 3 of 4 questions to level up!
• Probability of "at least one" success Get 3 of 4 questions to level up!

Solved Probability Problems

Solved probability problems and solutions are given here for a concept with clear understanding.

Students can get a fair idea on the probability questions which are provided with the detailed step-by-step answers to every question.

Solved probability problems with solutions :

The graphic above shows a container with 4 blue triangles, 5 green squares and 7 red circles. A single object is drawn at random from the container.

Match the following events with the corresponding probabilities:

Number of blue triangles in a container = 4

Number of green squares = 5

Number of red circles = 7

Total number of objects = 4 + 5 + 7 = 16

(i) The objects is not a circle:

P(the object is a circle)

= Number of circles/Total number of objects

P(the object is not a circle)

= 1 - P(the object is a circle)

= (16 - 7)/16

(ii) The objects is a triangle:

P(the object is a triangle)

= Number of triangle/Total number of objects

(iii) The objects is not a triangle:

= Number of triangles/Total number of objects

P(the object is not a triangle)

= 1 - P(the object is a triangle)

= (16 - 4)/16

(iv) The objects is not a square:

P(the object is a square)

= Number of squares/Total number of objects

P(the object is not a square)

= 1 - P(the object is a square)

= (16 - 5)/16

(v) The objects is a circle:

(vi) The objects is a square:

Match the following events with the corresponding probabilities are shown below:

2. A single card is drawn at random from a standard deck of 52 playing cards.

Match each event with its probability.

Note: fractional probabilities have been reduced to lowest terms. Consider the ace as the highest card.

Total number of playing cards = 52

(i) The card is a diamond:

Number of diamonds in a deck of 52 cards = 13

P(the card is a diamond)

= Number of diamonds/Total number of playing cards

(ii) The card is a red king:

Number of red king in a deck of 52 cards = 2

P(the card is a red king)

= Number of red kings/Total number of playing cards

(iii) The card is a king or queen:

Number of kings in a deck of 52 cards = 4

Number of queens in a deck of 52 cards = 4

Total number of king or queen in a deck of 52 cards = 4 + 4 = 8

P(the card is a king or queen)

= Number of king or queen/Total number of playing cards

(iv) The card is either a red card or an ace:

Total number of red card or an ace in a deck of 52 cards = 28

P(the card is either a red card or an ace)

= Number of cards which is either a red card or an ace/Total number of playing cards

(v) The card is not a king:

P(the card is a king)

= Number of kings/Total number of playing cards

P(the card is not a king)

= 1 - P(the card is a king)

= (13 - 1)/13

(vi) The card is a five or lower:

Number of cards is a five or lower = 16

P(the card is a five or lower)

= Number of card is a five or lower/Total number of playing cards

(vii) The card is a king:

(viii) The card is black:

Number of black cards in a deck of 52 cards = 26

P(the card is black)

= Number of black cards/Total number of playing cards

3. A bag contains 3 red balls and 4 black balls. A ball is drawn at random from the bag. Find the probability that the ball drawn is 

(ii) not black.

(i) Total number of possible outcomes = 3 + 4 = 7.

Number of favourable outcomes for the event E.

                              = Number of black balls = 4.

So, P(E) = \(\frac{\textrm{Number of Favourable Outcomes for the Event E}}{\textrm{Total Number of Possible Outcomes}}\)

             = \(\frac{4}{7}\).

(ii) The event of the ball being not black = \(\bar{E}\).

Hence, required probability = P(\(\bar{E}\))

                                        = 1 - P(E)

                                        = 1 - \(\frac{4}{7}\)

                                        = \(\frac{3}{7}\).

4. If the probability of Serena Williams a particular tennis match is 0.86, what is the probability of her losing the match?

Let E = the event of Serena Williams winning.

From the question, P(E) = 0.86.

Clearly, \(\bar{E}\) = the event of Serena Williams losing.

So, P(\(\bar{E}\)) = 1 - P(E) 

                            = 1 - 0.86

                            = 0.14

                            = \(\frac{14}{100}\)

                            = \(\frac{7}{50}\).

5. Find the probability of getting 53 Sunday in a leap year.

A leap year has 366 days. So, it has 52 weeks and 2 days.

So, 52 Sundays are assured. For 53 Sundays, one of the two remaining days must be a Sunday. 

For the remaining 2 days we can have

(Sunday, Monday), (Monday, Tuesday), (Tuesday, Wednesday), (Wednesday, Thursday), (Thursday, Friday), (Friday, Saturday), (Saturday, Sunday).

So, total number of possible outcomes = 7.

Number of favourable outcomes for the event E = 2,   [namely, (Sunday, Monday), (Saturday, Sunday)].

So, by definition: P(E) = \(\frac{2}{7}\).

6. A lot of 24 bulbs contains 25% defective bulbs. A bulb is drawn at random from the lot. It is found to be not defective and it is not put back. Now, one bulb is drawn at random from the rest. What is the probability that this bulb is not defective?

25% of 24 = \(\frac{25}{100}\) × 24 = 6.

So, there are 6 defective bulbs and 18 bulbs are not defective. 

After the first draw, the lot is left with 6 defective bulbs and 17 non-defective bulbs.

So, when the second bulb is drwn, the total number of possible outcomes = 23  (= 6+ 17).

Number of favourable outcomes for the event E = number of non-defective bulbs = 17.

So, the required probability = P(E) = (\frac{17}{23}\).

The examples can help the students to practice more questions on probability by following the concept provided in the solved probability problems.

• Probability

Random Experiments

Experimental Probability

Events in Probability

Empirical Probability

Coin Toss Probability

Probability of Tossing Two Coins

Probability of Tossing Three Coins

Complimentary Events

Mutually Exclusive Events

Mutually Non-Exclusive Events

Conditional Probability

Theoretical Probability

Odds and Probability

Playing Cards Probability

Probability and Playing Cards

Probability for Rolling Two Dice

Probability for Rolling Three Dice

• 9th Grade Math

From Solved Probability Problems to HOME PAGE

Didn't find what you were looking for? Or want to know more information about Math Only Math . Use this Google Search to find what you need.

New! Comments

• Preschool Activities
• Kindergarten Math
• 1st Grade Math
• 2nd Grade Math
• 3rd Grade Math
• 4th Grade Math
• 5th Grade Math
• 6th Grade Math
• 7th Grade Math
• 8th Grade Math
• 10th Grade Math
• 11 & 12 Grade Math
• Concepts of Sets
• Boolean Algebra
• Math Coloring Pages
• Multiplication Table
• Cool Maths Games
• Math Flash Cards
• Online Math Quiz
• Math Puzzles
• Binary System
• Math Dictionary
• Conversion Chart
• Homework Sheets
• Math Problem Ans
• Free Math Answers
• Printable Math Sheet
• Funny Math Answers
• Employment Test
• Math Patterns
• Link Partners
• Privacy Policy

Recent Articles

Area | Units to find Area | Conversion Table of Area | Two Dimensional

Apr 28, 24 10:14 AM

Math Only Math | Learn Math Step-by-Step | Worksheet | Videos | Games

Apr 27, 24 02:23 PM

Worksheet on Use of Decimal | Free Printable Decimals Worksheets

Apr 27, 24 01:45 PM

Adding 1-Digit Number | Understand the Concept one Digit Number

Apr 26, 24 01:55 PM

Subtracting 2-Digit Numbers | How to Subtract Two Digit Numbers?

Apr 26, 24 12:36 PM

© and ™ math-only-math.com. All Rights Reserved. 2010 - 2024.

• + ACCUPLACER Mathematics
• + ACT Mathematics
• + AFOQT Mathematics
• + ALEKS Tests
• + ASVAB Mathematics
• + ATI TEAS Math Tests
• + Common Core Math
• + DAT Math Tests
• + FSA Tests
• + FTCE Math
• + GED Mathematics
• + Georgia Milestones Assessment
• + GRE Quantitative Reasoning
• + HiSET Math Exam
• + HSPT Math
• + ISEE Mathematics
• + PARCC Tests
• + Praxis Math
• + PSAT Math Tests
• + PSSA Tests
• + SAT Math Tests
• + SBAC Tests
• + SIFT Math
• + SSAT Math Tests
• + STAAR Tests
• + TABE Tests
• + TASC Math
• + TSI Mathematics
• + ACT Math Worksheets
• + Accuplacer Math Worksheets
• + AFOQT Math Worksheets
• + ALEKS Math Worksheets
• + ASVAB Math Worksheets
• + ATI TEAS 6 Math Worksheets
• + FTCE General Math Worksheets
• + GED Math Worksheets
• + 3rd Grade Mathematics Worksheets
• + 4th Grade Mathematics Worksheets
• + 5th Grade Mathematics Worksheets
• + 6th Grade Math Worksheets
• + 7th Grade Mathematics Worksheets
• + 8th Grade Mathematics Worksheets
• + 9th Grade Math Worksheets
• + HiSET Math Worksheets
• + HSPT Math Worksheets
• + ISEE Middle-Level Math Worksheets
• + PERT Math Worksheets
• + Praxis Math Worksheets
• + PSAT Math Worksheets
• + SAT Math Worksheets
• + SIFT Math Worksheets
• + SSAT Middle Level Math Worksheets
• + 7th Grade STAAR Math Worksheets
• + 8th Grade STAAR Math Worksheets
• + THEA Math Worksheets
• + TABE Math Worksheets
• + TASC Math Worksheets
• + TSI Math Worksheets
• + AFOQT Math Course
• + ALEKS Math Course
• + ASVAB Math Course
• + ATI TEAS 6 Math Course
• + CHSPE Math Course
• + FTCE General Knowledge Course
• + GED Math Course
• + HiSET Math Course
• + HSPT Math Course
• + ISEE Upper Level Math Course
• + SHSAT Math Course
• + SSAT Upper-Level Math Course
• + PERT Math Course
• + Praxis Core Math Course
• + SIFT Math Course
• + 8th Grade STAAR Math Course
• + TABE Math Course
• + TASC Math Course
• + TSI Math Course
• + Number Properties Puzzles
• + Algebra Puzzles
• + Geometry Puzzles
• + Intelligent Math Puzzles
• + Ratio, Proportion & Percentages Puzzles
• + Other Math Puzzles

How to Solve Probability Problems? (+FREE Worksheet!)

Do you want to know how to solve Probability Problems? Here you learn how to solve probability word problems.

Related Topics

• How to Interpret Histogram
• How to Interpret Pie Graphs
• How to Solve Permutations and Combinations
• How to Find Mean, Median, Mode, and Range of the Given Data

Step by step guide to solve Probability Problems

• Probability is the likelihood of something happening in the future. It is expressed as a number between zero (can never happen) to \(1\) (will always happen).
• Probability can be expressed as a fraction, a decimal, or a percent.
• To solve a probability problem identify the event, find the number of outcomes of the event, then use probability law: \(\frac{number\ of \ favorable \ outcome}{total \ number \ of \ possible \ outcomes}\)

The Absolute Best Books to Ace Pre-Algebra to Algebra II

The Ultimate Algebra Bundle From Pre-Algebra to Algebra II

Probability problems – example 1:.

If there are \(8\) red balls and \(12\) blue balls in a basket, what is the probability that John will pick out a red ball from the basket?

There are \(8\) red balls and \(20\) a total number of balls. Therefore, the probability that John will pick out a red ball from the basket is \(8\) out of \(20\) or \(\frac{8}{8+12}=\frac{8}{20}=\frac{2}{5}\).

Probability Problems – Example 2:

A bag contains \(18\) balls: two green, five black, eight blue, a brown, a red, and one white. If \(17\) balls are removed from the bag at random, what is the probability that a brown ball has been removed?

If \(17\) balls are removed from the bag at random, there will be one ball in the bag. The probability of choosing a brown ball is \(1\) out of \(18\). Therefore, the probability of not choosing a brown ball is \(17\) out of \(18\) and the probability of having not a brown ball after removing \(17\) balls is the same.

Exercises for Solving Probability Problems

The Best Book to Help You Ace Pre-Algebra

Pre-Algebra for Beginners The Ultimate Step by Step Guide to Preparing for the Pre-Algebra Test

• A number is chosen at random from \(1\) to \(10\). Find the probability of selecting a \(4\) or smaller.
• A number is chosen at random from \(1\) to \(50\). Find the probability of selecting multiples of \(10\).
• A number is chosen at random from \(1\) to \(10\). Find the probability of selecting of \(4\) and factors of \(6\).
• A number is chosen at random from \(1\) to \(10\). Find the probability of selecting a multiple of \(3\).
• A number is chosen at random from \(1\) to \(50\). Find the probability of selecting prime numbers.
• A number is chosen at random from \(1\) to \(25\). Find the probability of not selecting a composite number.

Download Probability Problems Worksheet

• \(\color{blue}{\frac{2}{5}}\)
• \(\color{blue}{\frac{1}{10}}\)
• \(\color{blue}{\frac{1}{2}}\)
• \(\color{blue}{\frac{3}{10}}\)
• \(\color{blue}{\frac{9}{25}}\)

The Greatest Books for Students to Ace the Algebra

Pre-Algebra Exercise Book A Comprehensive Workbook + PreAlgebra Practice Tests

Pre-algebra in 10 days the most effective pre-algebra crash course, college algebra practice workbook the most comprehensive review of college algebra, high school algebra i a comprehensive review and step-by-step guide to mastering high school algebra 1, 10 full length clep college algebra practice tests the practice you need to ace the clep college algebra test.

by: Effortless Math Team about 4 years ago (category: Articles , Free Math Worksheets )

Effortless Math Team

Related to this article, more math articles.

• Discontinuous Function
• How to Multiply a Polynomial and a Monomial? (+FREE Worksheet!)
• How to Prepare for the Praxis Core Math Test?
• Meet the Key Reasons to Start Learning Math Now
• How to Create a GED Math Study Plan?
• How to Master Work Problems: A Comprehensive Step-by-Step Guide
• Integrals: Everything You Need To Know
• Top 10 Pre-Algebra Practice Questions
• 7th Grade WY-TOPP Math Worksheets: FREE & Printable
• The Ultimate Georgia Milestones Assessment Algebra 1 Course (+FREE Worksheets)

What people say about "How to Solve Probability Problems? (+FREE Worksheet!) - Effortless Math: We Help Students Learn to LOVE Mathematics"?

No one replied yet.

Leave a Reply Cancel reply

You must be logged in to post a comment.

Pre-Algebra Practice Workbook The Most Comprehensive Review of Pre-Algebra

Algebra i practice workbook the most comprehensive review of algebra 1, algebra ii practice workbook the most comprehensive review of algebra 2, algebra i for beginners the ultimate step by step guide to acing algebra i, algebra ii for beginners the ultimate step by step guide to acing algebra ii, pre-algebra tutor everything you need to help achieve an excellent score.

• ATI TEAS 6 Math
• ISEE Upper Level Math
• SSAT Upper-Level Math
• Praxis Core Math
• 8th Grade STAAR Math

Limited time only!

Save Over 45 %

It was $89.99 now it is $49.99

Login and use all of our services.

Effortless Math services are waiting for you. login faster!

Register Fast!

Password will be generated automatically and sent to your email.

After registration you can change your password if you want.

• Math Worksheets
• Math Courses
• Math Topics
• Math Puzzles
• Math eBooks
• GED Math Books
• HiSET Math Books
• ACT Math Books
• ISEE Math Books
• ACCUPLACER Books
• Premium Membership
• Youtube Videos

Effortless Math provides unofficial test prep products for a variety of tests and exams. All trademarks are property of their respective trademark owners.

• Bulk Orders
• Refund Policy

Browse Course Material

Course info, instructors.

• Dr. Jeremy Orloff
• Dr. Jennifer French Kamrin

Departments

• Mathematics

As Taught In

• Discrete Mathematics
• Probability and Statistics

Learning Resource Types

Introduction to probability and statistics, exams with solutions.

You are leaving MIT OpenCourseWare

Reset password New user? Sign up

Existing user? Log in

• Number Theory
• Probability
• Everyday Math
• Classical Mechanics
• Electricity and Magnetism
• Computer Science
• Quantitative Finance

Take a guided, problem-solving based approach to learning Probability. These compilations provide unique perspectives and applications you won't find anywhere else.

Probability Fundamentals

What's inside.

• Introduction
• Starting with Probability
• Principles of Probability
• Roll the Dice
• Fairness and Expected Value

Perplexing Probability

• Survive This Chapter
• Brain-Warping Probability
• Crazy Mad Dice

Casino Probability

• Casino Craps

Community Wiki

Browse through thousands of Probability wikis written by our community of experts.

• Rule of Sum
• Counting Integers in a Range
• Rule of Product
• Rule of Sum and Rule of Product Problem Solving
• Counting Shapes
• Discrete Mathematics
• Set Complement
• Permutations
• Permutations with Repetition
• Permutations With Restriction
• Combinatorics
• Combinations
• Rectangular Grid Walk

Sets and Partitions

• Sets - Elements
• Sets - Subsets
• Classical Sets
• Cardinality
• Partially Ordered Sets
• Ordinal Numbers
• Venn Diagram
• Union and Intersection
• Principle of Inclusion and Exclusion (PIE)
• Double Counting
• Derangements
• Bijection, Injection, And Surjection
• Bijective Functions
• Burnside's Lemma
• Pigeonhole Principle
• Ramsey Theory
• Dilworth's Theorem
• Distinct Objects into Distinct Bins
• Distinct Objects into Identical Bins
• Identical Objects into Distinct Bins
• Identical Objects into Identical Bins
• Partition of an Integer
• Mean (Average)
• Interquartile Range (IQR)
• Understanding Data - Problem Solving
• Data Presentation - Tables
• Data Presentation - Bar Charts
• Data Presentation - Pie Charts
• Data Presentation - Dot Diagram
• Data Presentation - Histogram
• Data Presentation - Stem-leaf plot
• Uniform Probability (by Outcomes)
• Probability - Rule of Sum
• Probability - Rule of Product
• Probability - By Complement
• Probability - Problem Solving
• Math of Poker - Basics
• Conditional Probability
• Probability - Independent events
• Probabilistic Principle of Inclusion and Exclusion
• Bayes' Theorem and Conditional Probability
• Conditional Probability Distribution
• Monty Hall Problem
• Bayesian Theory in Science and Math
• Poker Strategy
• Expected Value
• Linearity of Expectation
• Law of Iterated Expectation
• Coupon Collector Problem
• Standard Deviation
• Geometric Probability

Random Variables

• Discrete Random Variables - Definition
• Discrete Random Variables - Probability Density Function (PDF)
• Discrete Random Variables - Cumulative Distribution Function
• Discrete Random Variables - Joint Probability Distribution
• Discrete Random Variables - Indicator Variables
• Continuous Random Variables - Definition
• Continuous Random Variables - Probability Density Function (PDF)
• Continuous Random Variables - Cumulative Distribution Function
• Continuous Random Variables - Joint Probability Distribution
• Bernoulli Distribution
• Binomial Distribution
• Geometric Distribution
• Poisson Distribution
• Hypergeometric Distribution
• Continuous Probability Distributions - Uniform Distribution
• Normal Distribution
• Exponential Distribution
• Log-normal Distribution
• Multivariate Normal Distribution
• Sampling (Statistics)
• Chi-Squared Test
• Maximum Likelihood Estimation (MLE)
• Inverse Transform Sampling
• Statistical Significance
• Markov Chains
• Stationary Distributions of Markov Chains
• Transience and Recurrence of Markov Chains
• Ergodic Markov Chains
• Absorbing Markov Chains
• Binomial Theorem
• Binomial Coefficient
• Pascal's Triangle
• Negative Binomial Theorem
• Fractional Binomial Theorem
• Multinomial Coefficients
• Multinomial Theorem
• Recurrence Relations
• Minkowski Dimension
• Fibonacci Sequence
• Linear Recurrence Relations

Advanced Discrete

• Combinatorial Games - Definition
• Combinatorial Games - Winning Positions
• Sprague Grundy Theorem
• Dots and Boxes
• Fair Division
• Axiom of Determinacy
• Conjectures
• Construction
• Extremal Principle
• Invariant Principle
• Vandermonde's Identity
• Hockey Stick Identity
• Graph Theory
• Eulerian Path
• Hamiltonian Path
• Four Color Theorem
• Graph Coloring and Chromatic Numbers
• Hall's Marriage Theorem
• Applications of Hall's Marriage Theorem
• Art Gallery Problem
• Wiki Collaboration Graph
• Combinatorial Optimization
• Group actions
• Generating Functions
• Manipulating Generating Functions
• Coloring (Parity Arguments)

Problem Loading...

Note Loading...

Set Loading...

Probabilistic World

Probability Questions from the Real World (With Simulations)

Posted on August 2, 2020 Written by The Cthaeh Leave a Comment

Welcome to my introductory post to a large series that I’m starting today. The main purpose of this post is to get you in the mood for the posts to follow. Namely, exploring and solving interesting probability questions from the real world.

Most of my posts so far have been more on the theoretical side. In previous posts, I introduced important concepts from probability theory (and related fields like statistics and combinatorics):

• probabilities and sample spaces
• the law of large numbers and expected value
• permutations, combinations, and other combinatorics concepts
• mean and variance
• probability distributions
• Bayes’ theorem

And I’m going to continue introducing more concepts in the future, basic and advanced concepts alike. I personally find them fascinating in their own right. In the context of mathematics, they are interesting, thought provoking, and (some would say) even beautiful.

But they’re not just interesting, they’re also extremely useful. And when I say useful, I don’t just mean useful for mathematicians and scientists. I would argue they are a potentially useful tool for everybody.

Table of Contents

Skills in the real world

Think about the following skills for a moment:

• digesting nutrition
• absorbing water

You might find it strange that I’m calling these “skills”, but essentially they are. Of course, they are skills related to basic biological survival and, by definition, (almost) every living organism needs to have them in order to remain such.

Now, what about skills like:

• being fluent in a popular language
• detecting misinformation
• imagination and creativity
• performing CPR

In my opinion, skills like these (and many more) are always good to have. Regardless of your job, your age, or where you live. They are obviously not as essential as the previous category, but all of them are things that will allow you to achieve outcomes and take advantage of situations which you otherwise might not be able to.

Are probability theory skills useful in the real world?

So, where does having probability theory skills fit in all this? Well, I think it easily fits in the second category, though this isn’t as obviously true as some of the other skills in the list above. But think about it, what is probability theory really about? What does the ability to accurately calculate (or at least estimate) probabilities of events give you?

Well, probability theory is really about providing a measure for our uncertainty about an event’s occurrence and/or giving us insights about the frequency of an event’s long-term occurrence. In short, it helps us build good expectations about real-world events and phenomena. And, consequently, this helps us make better decisions (in the most general sense).

There’s uncertainty in so many fields. You can apply probability theory in science, games, economics, education, politics, and many more. Really, it’s hard to even come up with examples where probability theory can’t help. Regardless of what you do or find interesting, probability theory is a very useful tool to have under your belt.

Well, that’s how I feel about it anyway.

My motivation for these posts

Convincing you that probability theory is cool.

So, in an effort to justify my position, in this series I want to show you many probability questions from diverse areas in life. I’m going to start with simpler problems which are more fun than useful. And, eventually, I’m going to build up to more complicated ones.

More importantly, the process of solving these problems itself is useful in training your brain to think about probability questions. Often, the principles involved in solving simpler problems are the same (or at least similar) to the ones used for solving more complex ones.

Even though most of my posts so far have been theoretical, I’ve also written a few more practical ones. For example, I’ve shown you how to apply some of the theoretical concepts from the beginning for things like:

• solving the inverse problem
• calculating the bias of a coin
• predicting presidential elections
• cryptography
• Occam’s razor

But in more than one occasion I’ve been asked to give more examples of practical applications of the theoretical concepts, as well as just examples of solving probability related problems. Hopefully, this series will be a good first step in this direction.

Probability questions from the book Understanding Probability

People have also asked me for recommendations on probability theory and statistics books that give a decent overview of all important concepts from these fields.

For the first posts in this series, I’m going to use twelve probability questions from the book Understanding Probability: Chance Rules in Everyday Life by the author Henk Tijms . Tijms is a Dutch mathematician who specializes in probability theory and many related fields. If you’ve been interested in probability theory for long enough, this is a name that you’ve likely already heard.

I personally read this book a little less than 10 years ago while I was still finishing my master’s degree in cognitive neuroscience and back then I found it one of the most interesting books on the subject. I was pleasantly surprised when I recently received an email from Henk Tijms himself in which he shared some positive words about Probabilistic World. And he was kind enough to give me permission to use the probability questions from his book in my posts.

The very first image in this post (the funny laundry cartoon) is actually from the same book. It is the header image of the first chapter in which the twelve questions are introduced. I say introduced because the actual solutions are given in later chapters.

Anyway, if you’re new to probability theory and statistics and you’re looking for a good comprehensive book on the subject, I recommend you start with this book. Now, some of the concepts Henk Tijms discusses in the book are things that I’ve discussed myself. And the rest are things I’m going to discuss in the future. But when you read about the same concept explained in different ways by different people, this helps you consolidate your knowledge and understanding. This is an approach that I myself have used for a very long time and I find it very effective in learning.

So, I think the Understanding Probability book is a very good complement to my website.

Answering probability questions with simulations

My third main motive for this series is that I want to introduce you to the method of answering probability questions using simulations. This is an extremely important technique and sometimes it’s the only way certain questions can be answered. Why? Well, as you’ll see in future posts, there are a lot of problems for which we don’t have an analytic solution.

I’ve already used simulations in some of my previous posts:

• estimating coin bias
• the mean, the mode, and the median
• the law of large numbers
• expected value
• mean and variance of probability distributions

But, except for the first post in this list, I didn’t share the computer code used in these simulations. In order to show you how to use simulations yourself, in this series I’m going to be much more explicit with my explanations. And, for all simulations, I’m going to use my favorite programming language Python .

Python is an extremely powerful language and is one of the top choices for programmers, scientists, and basically anybody doing math-related programming for whatever reason. It’s also extremely beginner-friendly, easy to learn, and surprisingly similar to a natural language (English).

But don’t worry. If you don’t know anything about Python or programming in general, I’m going to make sure you still benefit from this series to the fullest extent. The simulations themselves are going to be ones you can perform even with a pen and paper. The role of the programming code is simply to make your computer perform the same steps automatically and much faster. Even without a programming background, you’ll gain intuition about the simulations.

For each probability question, I’m going to first show its analytic solution and then compare it to the answer we get with a simulation. Meaning, we’re going to reach the same answer from two entirely different paths. Which is going to be a very useful exercise for gaining intuition about the law of large numbers too!

What is a computer simulation?

In a nutshell, computer simulations are used for estimating probabilities empirically. This involves repeating the process that leads to the outcomes we’re interested in a large number of times. In the meantime, you simply keep track of the number of times each outcome occur. And the goal of the computer is to automate the steps in order to save you (lots of) time and effort.

In my post on the law of large numbers I showed you a few examples of such empirical estimates of probabilities. When it comes to the process of flipping a fair coin, the law guarantees that the percentage of flips that turn up “heads” will converge to the probability of “heads”. Click on the image below to see how the empirical estimate of the probability converges to the real probability as the number of simulated flips increases:

Click on the image to start/restart the animation.

Technically, you don’t need a computer for this. Just take a real coin or a real die and flip/roll it multiple times while keeping track of the outcomes. Of course, doing it like that is just extremely laborious (especially for more complicated processes), so it’s much better to use a computer simulation.

Bottom line is that, as long as you can simulate the outcome generating process with a computer (with programming or otherwise), you can empirically estimate the probability of any outcome. All thanks to the law of large numbers!

You know nothing about programming?

If you’ve never done any programming in your life but still want to run the simulations, you can do it. And I really mean that. You don’t have to first read a book about programming or Python. You don’t have to follow any online tutorials. None of that.

Don’t get me wrong, if you’re generally interested in getting into programming, you can do those things as well. But I’m a big fan of the philosophy called “learning by doing”. Especially for programming. If right now you’re thinking to yourself “Really? I can still run and understand the code even if know absolutely nothing about programming?”… Yes, trust me, you will be able to. And you’ll most likely start picking up programming concepts in the process, even if you don’t set this as an explicit goal.

For one thing, you’ll be able to run the code by simple copy/pasting even if you don’t understand it at all. But, like I said, Python is one of the most readable programming languages in existence and, even if you read the code as if you were reading plain English, you’ll still understand a lot. Especially combined with my brief explanations.

By the way, like I said earlier, even if you choose to skip the programming parts of my posts, you won’t lose anything from the analytic answers to the probability questions. But if you want to make your first steps in programming with actual probability questions, this is going to be a very good opportunity for you. And the only thing you’re going to need to start is Python itself.

Normally, you can simply download and install Python from the official Python website . But if you’re completely new to Python and/or programming, I strongly recommend installing it with the platform called Anaconda and using it with the web application called Jupyter Notebook that comes along with Anaconda.

Anaconda and Jupyter Notebook

You can download Anaconda from their official website . Definitely download the one with the latest Python 3 version (not Python 2) and just be careful to choose the right option for your operating system.

Once you download and install Anaconda, you can get familiar with it by following this quick guide . In particular, pay attention to the part about Jupyter Notebook . This is an awesome web application for running Python code (among other things) that is automatically installed when you install Anaconda.

Jupyter Notebook is an extremely popular tool among programmers working in fields like data science, machine learning, artificial intelligence, and many others intersecting with probability theory, statistics, and mathematics in general. It runs in your browser (the same one you’re currently reading this post from) and you’ll be able to run the code from my posts with it. Not only that, Anaconda will automatically install many popular Python packages that have a ton of useful functionality for the fields I mentioned.

If you want to get your hands dirty, you can take a look at this somewhat more extensive Jupyter Notebook tutorial . But you don’t have to read all these things at once, you can also do that when you start practicing with the code from my posts.

Bottom line, all you need to do to be able to start running the code form my posts is:

• Download and install Anaconda
• Learn how to run Jupyter Notebook from the command prompt (spoiler: the command is simply jupyter notebook )
• Optionally, go through the short tutorials I linked to

If you encounter any issues with these steps, let me know in the comments below and me or another reader will help you what that.

The probability questions

So, here are the titles of the twelve probability questions, as listed in the opening chapter of the book Understanding Probability:

• A birthday problem ( analytic solution and Python simulation )
• Probability of winning streaks
• A scratch-and-win lottery
• A lotto problem
• Hitting the jackpot
• Who is the murderer?
• A coincidence problem
• A sock problem
• A statistical test problem
• The best-choice problem
• The Monty Hall dilemma
• An offer you can’t refuse — or can you?

Many of these questions are famous problems in probability theory. But here Henk Tijms presents them in a fun and informal format. My posts won’t necessarily be in the same order, since answering some of these questions requires knowledge of concepts I haven’t talked about yet and I might put them on hold until I do.

Of course, these twelve questions are only a starting point. I’m going to write many other posts on other questions, some of them famous, some of them ones I came up with myself. And yet others which are simply interesting real-world questions that can be answered with probability theory.

Solver Title

Generating PDF...

• Pre Algebra Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Number Line Expanded Form Mean, Median & Mode
• Algebra Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums Interval Notation Pi (Product) Notation Induction Logical Sets Word Problems
• Pre Calculus Equations Inequalities Scientific Calculator Scientific Notation Arithmetics Complex Numbers Polar/Cartesian Simultaneous Equations System of Inequalities Polynomials Rationales Functions Arithmetic & Comp. Coordinate Geometry Plane Geometry Solid Geometry Conic Sections Trigonometry
• Calculus Derivatives Derivative Applications Limits Integrals Integral Applications Integral Approximation Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series Fourier Transform
• Functions Line Equations Functions Arithmetic & Comp. Conic Sections Transformation
• Linear Algebra Matrices Vectors
• Trigonometry Identities Proving Identities Trig Equations Trig Inequalities Evaluate Functions Simplify
• Statistics Mean Geometric Mean Quadratic Mean Average Median Mode Order Minimum Maximum Probability Mid-Range Range Standard Deviation Variance Lower Quartile Upper Quartile Interquartile Range Midhinge Standard Normal Distribution
• Physics Mechanics
• Chemistry Chemical Reactions Chemical Properties
• Finance Simple Interest Compound Interest Present Value Future Value
• Economics Point of Diminishing Return
• Conversions Roman Numerals Radical to Exponent Exponent to Radical To Fraction To Decimal To Mixed Number To Improper Fraction Radians to Degrees Degrees to Radians Hexadecimal Scientific Notation Distance Weight Time Volume
• Pre Algebra
• One-Step Addition
• One-Step Subtraction
• One-Step Multiplication
• One-Step Division
• One-Step Decimals
• Two-Step Integers
• Two-Step Add/Subtract
• Two-Step Multiply/Divide
• Two-Step Fractions
• Two-Step Decimals
• Multi-Step Integers
• Multi-Step with Parentheses
• Multi-Step Rational
• Multi-Step Fractions
• Multi-Step Decimals
• Solve by Factoring
• Completing the Square
• Quadratic Formula
• Biquadratic
• Logarithmic
• Exponential
• Rational Roots
• Floor/Ceiling
• Equation Given Roots
• Newton Raphson
• Substitution
• Elimination
• Cramer's Rule
• Gaussian Elimination
• System of Inequalities
• Perfect Squares
• Difference of Squares
• Difference of Cubes
• Sum of Cubes
• Polynomials
• Distributive Property
• FOIL method
• Perfect Cubes
• Binomial Expansion
• Negative Rule
• Product Rule
• Quotient Rule
• Expand Power Rule
• Fraction Exponent
• Exponent Rules
• Exponential Form
• Logarithmic Form
• Absolute Value
• Rational Number
• Powers of i
• Complex Form
• Partial Fractions
• Is Polynomial
• Leading Coefficient
• Leading Term
• Standard Form
• Complete the Square
• Synthetic Division
• Linear Factors
• Rationalize Denominator
• Rationalize Numerator
• Identify Type
• Convergence
• Interval Notation
• Pi (Product) Notation
• Boolean Algebra
• Truth Table
• Mutual Exclusive
• Cardinality
• Caretesian Product
• Age Problems
• Distance Problems
• Cost Problems
• Investment Problems
• Number Problems
• Percent Problems
• Addition/Subtraction
• Multiplication/Division
• Dice Problems
• Coin Problems
• Card Problems
• Pre Calculus
• Linear Algebra
• Trigonometry
• Conversions

Most Used Actions

Number line.

• A\:fair\:coin\:is\:tossed\:four\:times\:what\:is\:the\:probability\:of\:obtaining\:at\:least\:three\:heads
• What\:is\:the\:probability\:of\:rolling\:2\:standard\:dice\:which\:sum\:to\:11
• One\:card\:is\:selected\:at\:random\:from\:a\:deck\:of\:cards.\:Determine\:the\:probability\:that\:the\:card\:selected\:is\:a\:9?

probability-problems-calculator

• Middle School Math Solutions – Inequalities Calculator Next up in our Getting Started maths solutions series is help with another middle school algebra topic - solving...

Please add a message.

Message received. Thanks for the feedback.

Want Better Math Grades?

✅ Unlimited Solutions

✅ Step-by-Step Answers

✅ Available 24/7

➕ Free Bonuses ($1085 value!)

On this page

• Search IntMath
• Math interactives
• About (site info)
• Uses of Trignometry
• ASCIIMath input, KaTeX output
• ASCIIMath input, LaTeX and KaTeX output
• Send Math in emails
• Syntax for ASCIIMathML
• Math Display Experiments
• Scientific Notebook
• Math Problem Solver

Related Sections

Math Tutoring

Need help? Chat with a tutor anytime, 24/7.

Probability Problem Solver

This tool combines the power of mathematical computation engine that excels at solving mathematical formulas with the power of artificial intelligence large language models to parse and generate natural language answers. This creates a math problem solver that's more accurate than ChatGPT, more flexible than a math calculator, and provides answers faster than a human tutor.

Sign up for free here .

Problem Solver Subjects

Our math problem solver that lets you input a wide variety of probability math problems and it will provide a step by step answer. This math solver excels at math word problems as well as a wide range of math subjects.

• Math Word Problems
• Pre-Algebra
• Geometry Graphing
• Trigonometry
• Precalculus
• Finite Math
• Linear Algebra

Here are example math problems within each subject that can be input into the calculator and solved. This list is constanstly growing as functionality is added to the calculator.

Basic Math Solutions

Below are examples of basic math problems that can be solved.

• Long Arithmetic
• Rational Numbers
• Operations with Fractions
• Ratios, Proportions, Percents
• Measurement, Area, and Volume
• Factors, Fractions, and Exponents
• Unit Conversions
• Data Measurement and Statistics
• Points and Line Segments

Math Word Problem Solutions

Math word problems require interpreting what is being asked and simplifying that into a basic math equation. Once you have the equation you can then enter that into the problem solver as a basic math or algebra question to be correctly solved. Below are math word problem examples and their simplified forms.

Word Problem: Rachel has 17 apples. She gives some to Sarah. Sarah now has 8 apples. How many apples did Rachel give her?

Simplified Equation: 17 - x = 8

Word Problem: Rhonda has 12 marbles more than Douglas. Douglas has 6 marbles more than Bertha. Rhonda has twice as many marbles as Bertha has. How many marbles does Douglas have?

Variables: Rhonda's marbles is represented by (r), Douglas' marbles is represented by (d) and Bertha's marbles is represented by (b)

Simplified Equation: {r = d + 12, d = b + 6, r = 2 �� b}

Word Problem: if there are 40 cookies all together and Angela takes 10 and Brett takes 5 how many are left?

Simplified: 40 - 10 - 5

Pre-Algebra Solutions

Below are examples of Pre-Algebra math problems that can be solved.

• Variables, Expressions, and Integers
• Simplifying and Evaluating Expressions
• Solving Equations
• Multi-Step Equations and Inequalities
• Ratios, Proportions, and Percents
• Linear Equations and Inequalities

Algebra Solutions

Below are examples of Algebra math problems that can be solved.

• Algebra Concepts and Expressions
• Points, Lines, and Line Segments
• Simplifying Polynomials
• Factoring Polynomials
• Linear Equations
• Absolute Value Expressions and Equations
• Radical Expressions and Equations
• Systems of Equations
• Quadratic Equations
• Inequalities
• Complex Numbers and Vector Analysis
• Logarithmic Expressions and Equations
• Exponential Expressions and Equations
• Conic Sections
• Vector Spaces
• 3d Coordinate System
• Eigenvalues and Eigenvectors
• Linear Transformations
• Number Sets
• Analytic Geometry

Trigonometry Solutions

Below are examples of Trigonometry math problems that can be solved.

• Algebra Concepts and Expressions Review
• Right Triangle Trigonometry
• Radian Measure and Circular Functions
• Graphing Trigonometric Functions
• Simplifying Trigonometric Expressions
• Verifying Trigonometric Identities
• Solving Trigonometric Equations
• Complex Numbers
• Analytic Geometry in Polar Coordinates
• Exponential and Logarithmic Functions
• Vector Arithmetic

Precalculus Solutions

Below are examples of Precalculus math problems that can be solved.

• Operations on Functions
• Rational Expressions and Equations
• Polynomial and Rational Functions
• Analytic Trigonometry
• Sequences and Series
• Analytic Geometry in Rectangular Coordinates
• Limits and an Introduction to Calculus

Calculus Solutions

Below are examples of Calculus math problems that can be solved.

• Evaluating Limits
• Derivatives
• Applications of Differentiation
• Applications of Integration
• Techniques of Integration
• Parametric Equations and Polar Coordinates
• Differential Equations

Statistics Solutions

Below are examples of Statistics problems that can be solved.

• Algebra Review
• Average Descriptive Statistics
• Dispersion Statistics
• Probability
• Probability Distributions
• Frequency Distribution
• Normal Distributions
• t-Distributions
• Hypothesis Testing
• Estimation and Sample Size
• Correlation and Regression

Finite Math Solutions

Below are examples of Finite Math problems that can be solved.

• Polynomials and Expressions
• Equations and Inequalities
• Linear Functions and Points
• Systems of Linear Equations
• Mathematics of Finance
• Statistical Distributions

Linear Algebra Solutions

Below are examples of Linear Algebra math problems that can be solved.

• Introduction to Matrices
• Linear Independence and Combinations

Chemistry Solutions

Below are examples of Chemistry problems that can be solved.

• Unit Conversion
• Atomic Structure
• Molecules and Compounds
• Chemical Equations and Reactions
• Behavior of Gases
• Solutions and Concentrations

Physics Solutions

Below are examples of Physics math problems that can be solved.

• Static Equilibrium
• Dynamic Equilibrium
• Kinematics Equations
• Electricity
• Thermodymanics

Geometry Graphing Solutions

Below are examples of Geometry and graphing math problems that can be solved.

• Step By Step Graphing
• Linear Equations and Functions
• Polar Equations

Looking for the old Mathway Calculator? We've moved it to here .

This tool combines the power of mathematical computation engine that excels at solving mathematical formulas with the power of GPT large language models to parse and generate natural language. This creates math problem solver thats more accurate than ChatGPT, more flexible than a calculator, and faster answers than a human tutor. Learn More.

Tips, tricks, lessons, and tutoring to help reduce test anxiety and move to the top of the class.

Email Address Sign Up