probability homework

Statistics 110: Probability

Strategic Practice and Homework Problems

Actively solving practice problems is essential for learning probability. Strategic practice problems are organized by concept, to test and reinforce understanding of that concept.  Homework problems  usually do not say which concepts are involved, and often require combining several concepts. Each of the Strategic Practice documents here contains a set of strategic practice problems, solutions to those problems, a homework assignment, and solutions to the homework assignment. Also included here are the exercises from the  book that are marked with an s, and solutions to those exercises. 

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High school statistics

Unit 1: displaying a single quantitative variable, unit 2: analyzing a single quantitative variable, unit 3: two-way tables, unit 4: scatterplots, unit 5: study design, unit 6: probability, unit 7: probability distributions & expected value.

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5.3: Probability Rules- “And” and “Or”

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• Maxie Inigo, Jennifer Jameson, Kathryn Kozak, Maya Lanzetta, & Kim Sonier
• Coconino Community College

Learning Objectives

Students will be able to:

• Determine if two events are mutually exclusive and/or independent.
• Apply the "Or" rule to calculate the probability that either of two events occurs.
• Apply the "And" rule to calculate the probability that both of two events occurs.

Many probabilities in real life involve more than one event. If we draw a single card from a deck we might want to know the probability that it is either red or a jack. If we look at a group of students, we might want to know the probability that a single student has brown hair and blue eyes. When we combine two events we make a single event called a compound event . To create a compound event, we can use the word “and” or the word “or” to combine events. It is very important in probability to pay attention to the words “and” and “or” if they appear in a problem. The word “and” restricts the field of possible outcomes to only those outcomes that simultaneously describe all events. The word “or” broadens the field of possible outcomes to those that describe one or more events.

Example \(\PageIndex{1}\): Counting Students

Suppose a teacher wants to know the probability that a single student in her class of 30 students is taking either Art or English. She asks the class to raise their hands if they are taking Art and counts 13 hands. Then she asks the class to raise their hands if they are taking English and counts 21 hands. The teacher then calculates

\[P(\text{Art or English}) = \dfrac{13+21}{30} = \dfrac{33}{30} \nonumber\]

The teacher knows that this is wrong because probabilities must be between zero and one, inclusive. After thinking about it she remembers that nine students are taking both Art and English. These students raised their hands each time she counted, so the teacher counted them twice. When we calculate probabilities we have to be careful to count each outcome only once.

Mutually Exclusive Events

An experiment consists of drawing one card from a well shuffled deck of 52 cards. Consider the events E : the card is red, F : the card is a five, and G : the card is a spade. It is possible for a card to be both red and a five at the same time but it is not possible for a card to be both red and a spade at the same time. It would be easy to accidentally count a red five twice by mistake. It is not possible to count a red spade twice.

Definition: Mutually Exclusive

Two events are mutually exclusive if they have no outcomes in common.

Example \(\PageIndex{2}\): Mutually Exclusive with Dice

Two fair dice are tossed and different events are recorded. Let the events E , F and G be as follows:

• E = {the sum is five} = {(1, 4), (2, 3), (3, 2), (4, 1)}
• F = {both numbers are even} = {(2, 2), (2, 4), (2, 6), (4, 2), (4, 4), (4, 6), (6, 2), (6, 4), (6, 6)}
• G = {both numbers are less than five} = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4), (4,1), (4, 2), (4, 3), (4,4)}
• Are events E and F mutually exclusive?

Yes. E and F are mutually exclusive because they have no outcomes in common. It is not possible to add two even numbers to get a sum of five.

• Are events E and G mutually exclusive?

No. E and G are not mutually exclusive because they have some outcomes in common. The pairs (1, 4), (2, 3), (3, 2) and (4, 1) all have sums of 5 and both numbers are less than five.

• Are events F and G mutually exclusive?

No. F and G are not mutually exclusive because they have some outcomes in common. The pairs (2, 2), (2, 4), (4, 2) and (4, 4) all have two even numbers that are less than five.

Addition Rule for “Or” Probabilities

The addition rule for probabilities is used when the events are connected by the word “or”. Remember our teacher in Example \(\PageIndex{1}\) at the beginning of the section? She wanted to know the probability that her students were taking either art or English. Her problem was that she counted some students twice. She needed to add the number of students taking art to the number of students taking English and then subtract the number of students she counted twice. After dividing the result by the total number of students she will find the desired probability. The calculation is as follows:

\[ \begin{align*} P(\text{art or English}) &= \dfrac{\# \text{ taking art + } \# \text{ taking English - } \# \text{ taking both}}{\text{total number of students}} \\[4pt] &= \dfrac{13+21-9}{30} \\[4pt] &= \dfrac{25}{30} \approx {0.833} \end{align*}\]

The probability that a student is taking art or English is 0.833 or 83.3%.

When we calculate the probability for compound events connected by the word “or” we need to be careful not to count the same thing twice. If we want the probability of drawing a red card or a five we cannot count the red fives twice. If we want the probability a person is blonde-haired or blue-eyed we cannot count the blue-eyed blondes twice. The addition rule for probabilities adds the number of blonde-haired people to the number of blue-eyed people then subtracts the number of people we counted twice.

If A and B are any events then

\[P(A\, \text{or}\, B) = P(A) + P(B) – P(A \,\text{and}\, B).\]

If A and B are mutually exclusive events then \(P(A \,\text{and}\, B) = 0\), so then

\[P(A \, \text{or}\, B) = P(A) + P(B).\]

Example \(\PageIndex{3}\): Additional Rule for Drawing Cards

A single card is drawn from a well shuffled deck of 52 cards. Find the probability that the card is a club or a face card.

There are 13 cards that are clubs, 12 face cards (J, Q, K in each suit) and 3 face cards that are clubs.

\[ \begin{align*} P(\text{club or face card}) &= P(\text{club}) + P(\text{face card}) - P(\text{club and face card}) \\[4pt] &= \dfrac{13}{52} + \dfrac{12}{52} - \dfrac{3}{52} \\[4pt] &= \dfrac{22}{52} = \dfrac{11}{26} \approx {0.423} \end{align*}\]

The probability that the card is a club or a face card is approximately 0.423 or 42.3%.

A simple way to check this answer is to take the 52 card deck and count the number of physical cards that are either clubs or face cards. If you were to set aside all of the clubs and face cards in the deck, you would end up with the following:

{2 Clubs, 3 Clubs, 4 Clubs, 5 Clubs, 6 Clubs, 7 Clubs, 8 Clubs, 9 Clubs, 10 Clubs, J Clubs, Q Clubs, K Clubs, A Clubs, J Hearts, Q Hearts, K Hearts, J Spades, Q Spades, K Spades, J Diamonds, Q Diamonds, K Diamonds}

That is 22 cards out of the 52 card deck, which gives us a probably of: \[ \begin{align*} \dfrac{22}{52} = \dfrac{11}{26} \approx {0.423} \end{align*}\]

This confirms our earlier answer using the formal Addition Rule.

Example \(\PageIndex{4}\): Addition Rule for Tossing a Coin and Rolling a Die

An experiment consists of tossing a coin then rolling a die. Find the probability that the coin lands heads up or the number is five.

Let H represent heads up and T represent tails up. The sample space for this experiment is S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}.

• There are six ways the coin can land heads up, {H1, H2, H3, H4, H5, H6}.
• There are two ways the die can land on five, {H5, T5}.
• There is one way for the coin to land heads up and the die to land on five, {H5}.

\[ \begin{align*} P(\text{heads or five}) &= P(\text{heads}) + P(\text{five}) - P(\text{both heads and five}) \\[4pt] &= \dfrac{6}{12} + \dfrac{2}{12} - \dfrac{1}{12} \\[4pt] &= \dfrac{7}{12} = \approx {0.583} \end{align*}\]

The probability that the coin lands heads up or the number is five is approximately 0.583 or 58.3%.

Example \(\PageIndex{5}\): Addition Rule for Satisfaction of Car Buyers

Two hundred fifty people who recently purchased a car were questioned and the results are summarized in the following table.

Find the probability that a person bought a new car or was not satisfied.

\[\begin{align*} P(\text{new car or not satisfied}) &= P(\text{new car}) + P(\text{not satisfied}) - P(\text{new car and not satisfied}) \\[4pt] &= \dfrac{120}{250} + \dfrac{75}{250} - \dfrac{28}{250} = \dfrac{167}{250} \approx 0.668 \end{align*}\]

The probability that a person bought a new car or was not satisfied is approximately 0.668 or 66.8%.

Independent Events

Sometimes we need to calculate probabilities for compound events that are connected by the word “and.” Tossing a coin multiple times or rolling dice are independent events. Each time you toss a fair coin the probability of getting heads is ½. It does not matter what happened the last time you tossed the coin. It’s similar for dice. If you rolled double sixes last time that does not change the probability that you will roll double sixes this time. Drawing two cards without replacement is not an independent event. When you draw the first card and set it aside, the probability for the second card is now out of 51 cards not 52 cards.

Definition: Independent Events

Two events are independent events if the occurrence of one event has no effect on the probability of the occurrence of the other event.

Example \(\PageIndex{6}\): Determining When Events are Independent

Are these events independent?

a) A fair coin is tossed two times. The two events are (1) first toss is a head and (2) second toss is a head.

b) The two events (1) “It will rain tomorrow in Houston” and (2) “It will rain tomorrow in Galveston” (a city near Houston).

c) You draw a card from a deck, then draw a second card without replacing the first.

a) The probability that a head comes up on the second toss is \(\frac{1}{2}\) regardless of whether or not a head came up on the first toss, so these events are independent .

b) These events are not independent because it is more likely that it will rain in Galveston on days it rains in Houston than on days it does not.

c) The probability of the second card being red depends on whether the first card is red or not, so these events are not independent .

Multiplication Rule for “And” Probabilities: Independent Events

If events A and B are independent events, then \( P(\text{A and B}) = P(A) \cdot P(B)\).

Example \(\PageIndex{7}\): Independent Events for Tossing Coins

Suppose a fair coin is tossed four times. What is the probability that all four tosses land heads up?

The tosses of the coins are independent events. Knowing a head was tossed on the first trial does not change the probability of tossing a head on the second trial.

\(P(\text{four heads in a row}) = P(\text{1st heads and 2nd heads and 3rd heads and 4th heads})\)

\( = P(\text{1st heads}) \cdot P(\text{2nd heads}) \cdot P(\text{3rd heads}) \cdot P(\text{4th heads})\)

\( = \dfrac{1}{2} \cdot \dfrac{1}{2} \cdot \dfrac{1}{2} \cdot \dfrac{1}{2}\)

\( = \dfrac{1}{16}\)

The probability that all four tosses land heads up is \(\dfrac{1}{16}\).

Example \(\PageIndex{8}\): Independent Events for Drawing Marbles

A bag contains five red and four white marbles. A marble is drawn from the bag, its color recorded and the marble is returned to the bag. A second marble is then drawn. What is the probability that the first marble is red and the second marble is white?

Since the first marble is put back in the bag before the second marble is drawn these are independent events.

\[\begin{align*} P(\text{1st red and 2nd white}) &= P(\text{1st red}) \cdot P(\text{2nd white}) \\[4pt] &= \dfrac{5}{9} \cdot \dfrac{4}{9} = \dfrac{20}{81}\end{align*}\]

The probability that the first marble is red and the second marble is white is \(\dfrac{20}{81}\).

Example \(\PageIndex{9}\): Independent Events for Faulty Alarm Clocks

Abby has an important meeting in the morning. She sets three battery-powered alarm clocks just to be safe. If each alarm clock has a 0.03 probability of malfunctioning, what is the probability that all three alarm clocks fail at the same time?

Since the clocks are battery powered we can assume that one failing will have no effect on the operation of the other two clocks. The functioning of the clocks is independent.

\[\begin{align*} P(\text{all three fail}) &= P(\text{first fails}) \cdot P(\text{second fails})\cdot P(\text{third fails}) \\[4pt] &= (0.03)(0.03)(0.03) \\[4pt] &= 2.7 \times 10^{-5} \end{align*}\]

The probability that all three clocks will fail is approximately 0.000027 or 0.0027%. It is very unlikely that all three alarm clocks will fail.

At Least Once Rule for Independent Events

Many times we need to calculate the probability that an event will happen at least once in many trials. The calculation can get quite complicated if there are more than a couple of trials. Using the complement to calculate the probability can simplify the problem considerably. The following example will help you understand the formula.

Example \(\PageIndex{10}\): At Least Once Rule

The probability that a child forgets her homework on a given day is 0.15. What is the probability that she will forget her homework at least once in the next five days?

Assume that whether she forgets or not one day has no effect on whether she forgets or not the second day.

If P (forgets) = 0.15, then P (not forgets) = 0.85.

\[\begin{align*} P(\text{forgets at least once in 5 tries}) &= P(\text{forgets 1, 2, 3, 4 or 5 times in 5 tries}) \\[4pt] & = 1 - P(\text{forgets 0 times in 5 tries}) \\[4pt] &= 1 - P(\text{not forget}) \cdot P(\text{not forget}) \cdot P(\text{not forget}) \cdot P(\text{not forget}) \cdot P(\text{not forget}) \\[4pt] &= 1 - (0.85)(0.85)(0.85)(0.85)(0.85) \\[4pt] & = 1 - (0.85)^{5} = 0.556 \end{align*}\]

The probability that the child will forget her homework at least one day in the next five days is 0.556 or 55.6%

The idea in Example \(\PageIndex{9}\) can be generalized to get the At Least Once Rule.

Definition: At Least Once Rule

If an experiment is repeated n times, the n trials are independent and the probability of event A occurring one time is P(A) then the probability that A occurs at least one time is: \(P(\text{A occurs at least once in n trials}) = 1 - P(\overline{A})^{n}\)

Example \(\PageIndex{11}\): At Least Once Rule for Bird Watching

The probability of seeing a falcon near the lake during a day of bird watching is 0.21. What is the probability that a birdwatcher will see a falcon at least once in eight trips to the lake?

Let A be the event that he sees a falcon so P(A) = 0.21. Then, \(P(\overline{A}) = 1 - 0.21 = 0.79\).

\(P(\text{at least once in eight tries}) = 1 - P(\overline{A})^{8}\)

\( = 1 - (0.79)^{8}\)

\( = 1 - (0.152) = 0.848\)

The probability of seeing a falcon at least once in eight trips to the lake is approximately 0.848 or 84.8%.

Example \(\PageIndex{12}\): At Least Once Rule for Guessing on Multiple Choice Tests

A multiple choice test consists of six questions. Each question has four choices for answers, only one of which is correct. A student guesses on all six questions. What is the probability that he gets at least one answer correct?

Let A be the event that the answer to a question is correct. Since each question has four choices and only one correct choice, \(P(\text{correct}) = \dfrac{1}{4}\).

That means \(P(\text{not correct}) =1 - \dfrac{1}{4} = \dfrac{3}{4}\).

\[ \begin{align*} P(\text{at least one correct in six trials}) &= 1 - P(\text{not correct})^{6} \\[4pt] &= 1 - \left(\dfrac{3}{4}\right)^{6} \\[4pt] &= 1 - (0.178) = 0.822 \end{align*}\]

The probability that he gets at least one answer correct is 0.822 or 82.2%.

Probabilities from Two-Way Tables

Two-way tables can be used to define events and find their probabilities using two different approaches: intuitively or using the probability rules. We can calculate “and” and "or" probabilities by combining the data in relevant cells.

Example \(\PageIndex{13}\): Probabilities from a Two-Way Table

Continuation of Example \(\PageIndex{5}\):

A person is chosen at random. Find the probability that the person:

• bought a new car

\[\begin{align*} P(\text{new car}) &= \dfrac{\text{number of new car}}{\text{number of people}} \\[4pt] &= \dfrac{120}{250} = 0.480 = 48.0 \% \end{align*} \]

• was satisfied

\[\begin{align*} P(\text{satisfied}) &= \dfrac{\text{number of satisfied}}{\text{number of people}} \\[4pt] &= \dfrac{175}{250} = 0.700 = 70.0 \% \end{align*} \]

• bought a new car and was satisfied

\[\begin{align*} P(\text{new car and satisfied}) &= \dfrac{\text{number of new car and satisfied}}{\text{number of people}} \\[4pt] &= \dfrac{92}{250} = 0.368 = 36.8 \% \end{align*} \]

• bought a new car or was satisfied

\[\begin{align*} P(\text{new car or satisfied}) &= \dfrac{\text{number of new car + number of satisfied - number of new car and satisfied}}{\text{number of people}} \\[4pt] &= \dfrac{120 + 175 - 92}{250} = \dfrac{203}{250} = 0.812 = 81.2 \% \end{align*} \]

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Probability Worksheets

Navigate through this assortment of printable probability worksheets that includes exercises on basic probability based on more likely, less likely, equally likely, certain and impossible events, pdf worksheets based on identifying suitable events, simple spinner problems, for students in grade 4, grade 5, and grade 6. With the required introduction, the beginners get to further their knowledge with skills like probability on single coin, two coins, days in a week, months in a year, fair die, pair of dice, deck of cards, numbers and more. Mutually exclusive and inclusive events, probability on odds and other challenging probability worksheets are useful for grade 7, grade 8, and high school. Grab some of these probability worksheets for free!

Probability on Coins

Simple probability worksheets based on tossing single coin or two coins. Identify the proper sample space before finding probability.

Probability in a single coin toss

Probability in pair of coin - 1

Probability in pair of coin - 2

Probability on Days and Months

Fun filled worksheet pdfs based on days in a week and months in a year. Sample space is easy to find but care is required in identifying like events.

Days of a week

Months of a year - 1

Months of a year - 2

Probability on Fair Die

Fair die is numbered from 1 to 6. Understand the multiples, divisors and factors and apply it on these probability worksheets.

Simple numbers

Multiples and divisors

Mutually exclusive and inclusive

Probability on Pair of Dice

Sample space is little large which contains 36 elements. Write all of them in papers before start answering on probability questions for grade 7 and grade 8.

Based on numbers

Based on sum and difference

Based on multiples and divisors

Based on factors

Probability on Numbers

Students should learn the concepts of multiples, divisors and factors before start practicing these printable worksheets.

Probability on numbers - 1

Probability on numbers - 2

Probability on numbers - 3

Probability on numbers - 4

Probability on numbers - 5

Probability on Deck of Cards

Deck of cards contain 52 cards, 26 are black, 26 are red, four different flowers, each flower contain 13 cards such as A, 1, 2, ..., 10, J, Q, K.

Deck of cards worksheet - 1

Deck of cards worksheet - 2

Deck of cards worksheet - 3

Probability on Spinners

Interactive worksheets for 4th grade and 5th grade kids to understand the probability using spinners. Colorful spinners are included for more fun.

Spinner worksheets on numbers

Spinner worksheets on colors

Probability on Odds

Probability on odds worksheets can be broadly classifieds as favorable to the events or against the events.

Odds worksheet - 1

Odds worksheet - 2

Odds worksheet - 3

Probability on Independent and Dependent

Here comes our challenging probability worksheets set for 8th grade and high school students based on dependent and independent events with various real-life applications.

Based on deck of cards

Based on marbles

Based on cards

Probability on Different Events

Basic probability worksheets for beginners in 6th grade and 7th grade to understand the different type of events such as more likely, less likely, equally likely and so on.

Balls in container

Identify suitable events

Mutually inclusive and exclusive events

Related Worksheets

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Chapter Review

3.1 terminology.

In this module we learned the basic terminology of probability. The set of all possible outcomes of an experiment is called the sample space. Events are subsets of the sample space, and they are assigned a probability that is a number between zero and one, inclusive.

3.2 Independent and Mutually Exclusive Events

Two events A and B are independent if the knowledge that one occurred does not affect the chance the other occurs. If two events are not independent, then we say that they are dependent

In sampling with replacement, each member of a population is replaced after it is picked, so that member has the possibility of being chosen more than once, and the events are considered to be independent. In sampling without replacement, each member of a population may be chosen only once, and the events are considered not to be independent. When events do not share outcomes, they are mutually exclusive of each other.

3.3 Two Basic Rules of Probability

The multiplication rule and the addition rule are used for computing the probability of A and B , as well as the probability of A or B for two given events A , B defined on the sample space. In sampling with replacement, each member of a population is replaced after it is picked, so that member has the possibility of being chosen more than once, and the events are considered to be independent. In sampling without replacement, each member of a population may be chosen only once, and the events are considered to be not independent. The events A and B are mutually exclusive events when they do not have any outcomes in common.

3.4 Contingency Tables

There are several tools you can use to help organize and sort data when calculating probabilities. Contingency tables, also known as two-way tables, help display data and are particularly useful when calculating probabilites that have multiple dependent variables.

3.5 Tree and Venn Diagrams

A tree diagram uses branches to show the different outcomes of experiments and makes complex probability questions easy to visualize.

A Venn diagram is a picture that represents the outcomes of an experiment. It generally consists of a box that represents the sample space S together with circles or ovals. The circles or ovals represent events. A Venn diagram is especially helpful for visualizing the OR event, the AND event, and the complement of an event and for understanding conditional probabilities.

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3.H: Probability (Homework)

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3.1 Terminology

The graph in Figure \(\PageIndex{17}\) displays the sample sizes and percentages of people in different age and gender groups who were polled concerning their approval of Mayor Ford’s actions in office. The total number in the sample of all the age groups is 1,045.

• Define three events in the graph.
• Describe in words what the entry 40 means.
• Describe in words the complement of the entry in question 2.
• Describe in words what the entry 30 means.
• Out of the males and females, what percent are males?
• Out of the females, what percent disapprove of Mayor Ford?
• Out of all the age groups, what percent approve of Mayor Ford?
• Find P(Approve|Male).
• Out of the age groups, what percent are more than 44 years old?
• Find P(Approve|Age < 35).

Explain what is wrong with the following statements. Use complete sentences.

• If there is a 60% chance of rain on Saturday and a 70% chance of rain on Sunday, then there is a 130% chance of rain over the weekend.
• The probability that a baseball player hits a home run is greater than the probability that he gets a successful hit.

3.2 Independent and Mutually Exclusive Events

Use the following information to answer the next 12 exercises. The graph shown is based on more than 170,000 interviews done by Gallup that took place from January through December 2012. The sample consists of employed Americans 18 years of age or older. The Emotional Health Index Scores are the sample space. We randomly sample one Emotional Health Index Score.

Find the probability that an Emotional Health Index Score is 82.7.

Find the probability that an Emotional Health Index Score is 81.0.

Find the probability that an Emotional Health Index Score is more than 81?

Find the probability that an Emotional Health Index Score is between 80.5 and 82?

If we know an Emotional Health Index Score is 81.5 or more, what is the probability that it is 82.7?

What is the probability that an Emotional Health Index Score is 80.7 or 82.7?

What is the probability that an Emotional Health Index Score is less than 80.2 given that it is already less than 81.

What occupation has the highest emotional index score?

What occupation has the lowest emotional index score?

What is the range of the data?

Compute the average EHIS.

If all occupations are equally likely for a certain individual, what is the probability that he or she will have an occupation with lower than average EHIS?

3.3 Two Basic Rules of Probability

On February 28, 2013, a Field Poll Survey reported that 61% of California registered voters approved of allowing two people of the same gender to marry and have regular marriage laws apply to them. Among 18 to 39 year olds (California registered voters), the approval rating was 78%. Six in ten California registered voters said that the upcoming Supreme Court’s ruling about the constitutionality of California’s Proposition 8 was either very or somewhat important to them. Out of those CA registered voters who support same-sex marriage, 75% say the ruling is important to them.

In this problem, let:

• C = California registered voters who support same-sex marriage.
• B = California registered voters who say the Supreme Court’s ruling about the constitutionality of California’s Proposition 8 is very or somewhat important to them
• A = California registered voters who are 18 to 39 years old.
• Find \(P(C)\).
• Find \(P(B)\).
• Find \(P(C|A)\).
• Find \(P(B|C)\).
• In words, what is \(C|A\)?
• In words, what is \(B|C\)?
• Find \(P(C \cap B)\).
• In words, what is \(C \cap B\)?
• Find \(P(C \cup B)\).
• Are C and B mutually exclusive events? Show why or why not.

After Rob Ford, the mayor of Toronto, announced his plans to cut budget costs in late 2011, the Forum Research polled 1,046 people to measure the mayor’s popularity. Everyone polled expressed either approval or disapproval. These are the results their poll produced:

• In early 2011, 60 percent of the population approved of Mayor Ford’s actions in office.
• In mid-2011, 57 percent of the population approved of his actions.
• In late 2011, the percentage of popular approval was measured at 42 percent.
• What is the sample size for this study?
• What proportion in the poll disapproved of Mayor Ford, according to the results from late 2011?
• How many people polled responded that they approved of Mayor Ford in late 2011?
• What is the probability that a person supported Mayor Ford, based on the data collected in mid-2011?
• What is the probability that a person supported Mayor Ford, based on the data collected in early 2011?

Use the following information to answer the next three exercises. The casino game, roulette, allows the gambler to bet on the probability of a ball, which spins in the roulette wheel, landing on a particular color, number, or range of numbers. The table used to place bets contains of 38 numbers, and each number is assigned to a color and a range.

• List the sample space of the 38 possible outcomes in roulette.
• You bet on red. Find P(red).
• You bet on -1st 12- (1st Dozen). Find P(-1st 12-).
• You bet on an even number. Find P(even number).
• Is getting an odd number the complement of getting an even number? Why?
• Find two mutually exclusive events.
• Are the events Even and 1st Dozen independent?

Compute the probability of winning the following types of bets:

• Betting on two lines that touch each other on the table as in 1-2-3-4-5-6
• Betting on three numbers in a line, as in 1-2-3
• Betting on one number
• Betting on four numbers that touch each other to form a square, as in 10-11-13-14
• Betting on two numbers that touch each other on the table, as in 10-11 or 10-13
• Betting on 0-00-1-2-3
• Betting on 0-1-2; or 0-00-2; or 00-2-3
• Betting on a color
• Betting on one of the dozen groups
• Betting on the range of numbers from 1 to 18
• Betting on the range of numbers 19–36
• Betting on one of the columns
• Betting on an even or odd number (excluding zero)

Suppose that you have eight cards. Five are green and three are yellow. The five green cards are numbered 1, 2, 3, 4, and 5. The three yellow cards are numbered 1, 2, and 3. The cards are well shuffled. You randomly draw one card.

• G = card drawn is green
• List the sample space.
• \(P(G) =\) _____
• \(P(G|E) =\) _____
• \(P(G \cap E) =\) _____
• \(P(G \cup E) =\) _____
• Are G and E mutually exclusive? Justify your answer numerically.

Roll two fair dice separately. Each die has six faces.

• Let A be the event that either a three or four is rolled first, followed by an even number. Find \(P(A)\).
• Let B be the event that the sum of the two rolls is at most seven. Find \(P(B)\).
• In words, explain what “\(P(A|B)\)” represents. Find \(P(A|B)\).
• Are A and B mutually exclusive events? Explain your answer in one to three complete sentences, including numerical justification.
• Are A and B independent events? Explain your answer in one to three complete sentences, including numerical justification.

A special deck of cards has ten cards. Four are green, three are blue, and three are red. When a card is picked, its color of it is recorded. An experiment consists of first picking a card and then tossing a coin.

• Let A be the event that a blue card is picked first, followed by landing a head on the coin toss. Find P(A).
• Let B be the event that a red or green is picked, followed by landing a head on the coin toss. Are the events A and B mutually exclusive? Explain your answer in one to three complete sentences, including numerical justification.
• Let C be the event that a red or blue is picked, followed by landing a head on the coin toss. Are the events A and C mutually exclusive? Explain your answer in one to three complete sentences, including numerical justification.

An experiment consists of first rolling a die and then tossing a coin.

• Let A be the event that either a three or a four is rolled first, followed by landing a head on the coin toss. Find P(A).
• Let B be the event that the first and second tosses land on heads. Are the events A and B mutually exclusive? Explain your answer in one to three complete sentences, including numerical justification.

An experiment consists of tossing a nickel, a dime, and a quarter. Of interest is the side the coin lands on.

• Let A be the event that there are at least two tails. Find P(A).
• Let B be the event that the first and second tosses land on heads. Are the events A and B mutually exclusive? Explain your answer in one to three complete sentences, including justification.

Consider the following scenario: Let \(P(C) = 0.4\). Let \(P(D) = 0.5\). Let \(P(C|D) = 0.6\).

• Find \(P(C \cap D)\).
• Are C and D mutually exclusive? Why or why not?
• Are C and D independent events? Why or why not?
• Find \(P(C \cup D)\).
• Find \(P(D|C)\).

Y and Z are independent events.

• Rewrite the basic Addition Rule \(P(Y \cup Z) = P(Y) + P(Z) - P(Y \cap Z)\) using the information that Y and Z are independent events.
• Use the rewritten rule to find \(P(Z)\) if \(P(Y \cup Z) = 0.71\) and \(P(Y) = 0.42\).

G and H are mutually exclusive events. \(P(G) = 0.5 P(H) = 0.3\)

• Explain why the following statement MUST be false: \(P(H|G) = 0.4\).
• Find \(P(H \cup G)\).
• Are G and H independent or dependent events? Explain in a complete sentence.

Approximately 281,000,000 people over age five live in the United States. Of these people, 55,000,000 speak a language other than English at home. Of those who speak another language at home, 62.3% speak Spanish.

Let: E = speaks English at home; E′ = speaks another language at home; S = speaks Spanish;

Finish each probability statement by matching the correct answer.

1994, the U.S. government held a lottery to issue 55,000 Green Cards (permits for non-citizens to work legally in the U.S.). Renate Deutsch, from Germany, was one of approximately 6.5 million people who entered this lottery. Let G = won green card.

• What was Renate’s chance of winning a Green Card? Write your answer as a probability statement.
• In the summer of 1994, Renate received a letter stating she was one of 110,000 finalists chosen. Once the finalists were chosen, assuming that each finalist had an equal chance to win, what was Renate’s chance of winning a Green Card? Write your answer as a conditional probability statement. Let F = was a finalist.
• Are G and F independent or dependent events? Justify your answer numerically and also explain why.
• Are G and F mutually exclusive events? Justify your answer numerically and explain why.

Three professors at George Washington University did an experiment to determine if economists are more selfish than other people. They dropped 64 stamped, addressed envelopes with $10 cash in different classrooms on the George Washington campus. 44% were returned overall. From the economics classes 56% of the envelopes were returned. From the business, psychology, and history classes 31% were returned.

Let: R = money returned; E = economics classes; O = other classes

• Write a probability statement for the overall percent of money returned.
• Write a probability statement for the percent of money returned out of the economics classes.
• Write a probability statement for the percent of money returned out of the other classes.
• Is money being returned independent of the class? Justify your answer numerically and explain it.
• Based upon this study, do you think that economists are more selfish than other people? Explain why or why not. Include numbers to justify your answer.

The following table of data obtained from www.baseball-almanac.com shows hit information for four players. Suppose that one hit from the table is randomly selected.

Are "the hit being made by Hank Aaron" and "the hit being a double" independent events?

• Yes, because P(hit by Hank Aaron|hit is a double) = P(hit by Hank Aaron)
• No, because P(hit by Hank Aaron|hit is a double) ≠ P(hit is a double)
• No, because P(hit is by Hank Aaron|hit is a double) ≠ P(hit by Hank Aaron)
• Yes, because P(hit is by Hank Aaron|hit is a double) = P(hit is a double)

United Blood Services is a blood bank that serves more than 500 hospitals in 18 states. According to their website, a person with type O blood and a negative Rh factor (Rh-) can donate blood to any person with any bloodtype. Their data show that 43% of people have type O blood and 15% of people have Rh- factor; 52% of people have type O or Rh- factor.

• Find the probability that a person has both type O blood and the Rh- factor.
• Find the probability that a person does NOT have both type O blood and the Rh- factor.

At a college, 72% of courses have final exams and 46% of courses require research papers. Suppose that 32% of courses have a research paper and a final exam. Let F be the event that a course has a final exam. Let R be the event that a course requires a research paper.

• Find the probability that a course has a final exam or a research project.
• Find the probability that a course has NEITHER of these two requirements.

In a box of assorted cookies, 36% contain chocolate and 12% contain nuts. Of those, 8% contain both chocolate and nuts. Sean is allergic to both chocolate and nuts.

• Find the probability that a cookie contains chocolate or nuts (he can't eat it).
• Find the probability that a cookie does not contain chocolate or nuts (he can eat it).

A college finds that 10% of students have taken a distance learning class and that 40% of students are part time students. Of the part time students, 20% have taken a distance learning class. Let D = event that a student takes a distance learning class andE = event that a student is a part time student

• Find \(P(D \cap E)\).
• Find \(P(E|D)\).
• Find \(P(D \cup E)\).
• Using an appropriate test, show whether D and E are independent.
• Using an appropriate test, show whether D and E are mutually exclusive.

3.5 Venn Diagrams

Use the information in the Table \(\PageIndex{16}\) to answer the next eight exercises. The table shows the political party affiliation of each of 67 members of the US Senate in June 2012, and when they are up for reelection.

What is the probability that a randomly selected senator has an “Other” affiliation?

What is the probability that a randomly selected senator is up for reelection in November 2016?

What is the probability that a randomly selected senator is a Democrat and up for reelection in November 2016?

What is the probability that a randomly selected senator is a Republican or is up for reelection in November 2014?

Suppose that a member of the US Senate is randomly selected. Given that the randomly selected senator is up for reelection in November 2016, what is the probability that this senator is a Democrat?

Suppose that a member of the US Senate is randomly selected. What is the probability that the senator is up for reelection in November 2014, knowing that this senator is a Republican?

The events “Republican” and “Up for reelection in 2016” are ________

• mutually exclusive.
• independent.
• both mutually exclusive and independent.
• neither mutually exclusive nor independent.

The events “Other” and “Up for reelection in November 2016” are ________

Table \(\PageIndex{17}\) gives the number of participants in the recent National Health Interview Survey who had been treated for cancer in the previous 12 months. The results are sorted by age, race (black or white), and sex. We are interested in possible relationships between age, race, and sex. We will let suicide victims be our population.

Do not include "all others" for parts f and g.

• Fill in the column for cancer treatment for individuals over age 65.
• Fill in the row for all other races.
• Find the probability that a randomly selected individual was a white male.
• Find the probability that a randomly selected individual was a black female.
• Find the probability that a randomly selected individual was black
• Find the probability that a randomly selected individual was male.
• Out of the individuals over age 65, find the probability that a randomly selected individual was a black or white male.

Use the following information to answer the next two exercises. The table of data obtained from www.baseball-almanac.com shows hit information for four well known baseball players. Suppose that one hit from the table is randomly selected.

Find P(hit was made by Babe Ruth).

• \(\frac{1518}{2873}\)
• \(\frac{2873}{12351}\)
• \(\frac{583}{12351}\)
• \(\frac{4189}{12351}\)

Find P(hit was made by Ty Cobb|The hit was a Home Run).

• \(\frac{114}{1720}\)
• \(\frac{1720}{4189}\)
• \(\frac{114}{12351}\)

Table \(\PageIndex{19}\) identifies a group of children by one of four hair colors, and by type of hair.

• Complete the table.
• What is the probability that a randomly selected child will have wavy hair?
• What is the probability that a randomly selected child will have either brown or blond hair?
• What is the probability that a randomly selected child will have wavy brown hair?
• What is the probability that a randomly selected child will have red hair, given that he or she has straight hair?
• If B is the event of a child having brown hair, find the probability of the complement of B.
• In words, what does the complement of B represent?

In a previous year, the weights of the members of the San Francisco 49ers and the Dallas Cowboys were published in the San Jose Mercury News . The factual data were compiled into the following table.

For the following, suppose that you randomly select one player from the 49ers or Cowboys.

• Find the probability that his shirt number is from 1 to 33.
• Find the probability that he weighs at most 210 pounds.
• Find the probability that his shirt number is from 1 to 33 AND he weighs at most 210 pounds.
• Find the probability that his shirt number is from 1 to 33 OR he weighs at most 210 pounds.
• Find the probability that his shirt number is from 1 to 33 GIVEN that he weighs at most 210 pounds.

Use the following information to answer the next two exercises. This tree diagram shows the tossing of an unfair coin followed by drawing one bead from a cup containing three red (R), four yellow (Y) and five blue (B) beads. For the coin, P(H) = \(\frac{2}{3}\) and P(T) = \(\frac{1}{3}\) where H is heads and T is tails.

Find P(tossing a Head on the coin AND a Red bead)

• \(\frac{2}{3}\)
• \(\frac{5}{15}\)
• \(\frac{6}{36}\)
• \(\frac{5}{36}\)

Find P(Blue bead).

• \(\frac{15}{36}\)
• \(\frac{10}{36}\)
• \(\frac{10}{12}\)

A box of cookies contains three chocolate and seven butter cookies. Miguel randomly selects a cookie and eats it. Then he randomly selects another cookie and eats it. (How many cookies did he take?)

• Draw the tree that represents the possibilities for the cookie selections. Write the probabilities along each branch of the tree.
• Are the probabilities for the flavor of the SECOND cookie that Miguel selects independent of his first selection? Explain.
• For each complete path through the tree, write the event it represents and find the probabilities.
• Let S be the event that both cookies selected were the same flavor. Find P(S).
• Let T be the event that the cookies selected were different flavors. Find P(T) by two different methods: by using the complement rule and by using the branches of the tree. Your answers should be the same with both methods.
• Let U be the event that the second cookie selected is a butter cookie. Find P(U).

Probability

How likely something is to happen.

Many events can't be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.

Tossing a Coin

When a coin is tossed, there are two possible outcomes:

Heads (H) or Tails (T)

• the probability of the coin landing H is ½
• the probability of the coin landing T is ½

Throwing Dice

When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6 .

The probability of any one of them is 1 6

In general:

Probability of an event happening = Number of ways it can happen Total number of outcomes

Example: the chances of rolling a "4" with a die

Number of ways it can happen: 1 (there is only 1 face with a "4" on it)

Total number of outcomes: 6 (there are 6 faces altogether)

So the probability = 1 6

Example: there are 5 marbles in a bag: 4 are blue, and 1 is red. What is the probability that a blue marble gets picked?

Number of ways it can happen: 4 (there are 4 blues)

Total number of outcomes: 5 (there are 5 marbles in total)

So the probability = 4 5 = 0.8

Probability Line

We can show probability on a Probability Line :

Probability is always between 0 and 1

Probability is Just a Guide

Probability does not tell us exactly what will happen, it is just a guide

Example: toss a coin 100 times, how many Heads will come up?

Probability says that heads have a ½ chance, so we can expect 50 Heads .

But when we actually try it we might get 48 heads, or 55 heads ... or anything really, but in most cases it will be a number near 50.

Learn more at Probability Index .

Some words have special meaning in Probability:

Experiment : a repeatable procedure with a set of possible results.

Example: Throwing dice

We can throw the dice again and again, so it is repeatable.

The set of possible results from any single throw is {1, 2, 3, 4, 5, 6}

Outcome: A possible result.

Example: "6" is one of the outcomes of a throw of a die.

Trial: A single performance of an experiment.

Example: I conducted a coin toss experiment. After 4 trials I got these results:

Three trials had the outcome "Head", and one trial had the outcome "Tail"

Sample Space: all the possible outcomes of an experiment.

Example: choosing a card from a deck

There are 52 cards in a deck (not including Jokers)

So the Sample Space is all 52 possible cards : {Ace of Hearts, 2 of Hearts, etc... }

The Sample Space is made up of Sample Points:

Sample Point: just one of the possible outcomes

Example: Deck of Cards

• the 5 of Clubs is a sample point
• the King of Hearts is a sample point

"King" is not a sample point. There are 4 Kings, so that is 4 different sample points.

There are 6 different sample points in that sample space.

Event: one or more outcomes of an experiment

Example Events:

An event can be just one outcome:

• Getting a Tail when tossing a coin
• Rolling a "5"

An event can include more than one outcome:

• Choosing a "King" from a deck of cards (any of the 4 Kings)
• Rolling an "even number" (2, 4 or 6)

Hey, let's use those words, so you get used to them:

Example: Alex wants to see how many times a "double" comes up when throwing 2 dice.

The Sample Space is all possible Outcomes (36 Sample Points):

{1,1} {1,2} {1,3} {1,4} ... ... ... {6,3} {6,4} {6,5} {6,6}

The Event Alex is looking for is a "double", where both dice have the same number. It is made up of these 6 Sample Points :

{1,1} {2,2} {3,3} {4,4} {5,5} and {6,6}

These are Alex's Results:

 After 100 Trials , Alex has 19 "double" Events ... is that close to what you would expect?

Fall 2020, Lecture B00 (Kemp) TR 12:30-1:50pm

Probability theory i.

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Announcements

Last Modified: 9/23/15

• Welcome : to Math 280A: Probability Theory I, in Fall 2020!
• Quiz 1 grades have been released on Gradescope. The regrade request window is now close.
• Quiz 2 grades have been released on Gradescope. The regrade request window is Monday 10/26/20 8am to Wednesday 10/28/20 at 8pm . If you wish to lodge a regrade request on Quiz 1, please do so during this period of time.

Course Information

• Textbook : The main source we will follow are Bruce Driver's excellent Probability notes: Probability Tools with Examples , by Bruce Driver Here are a few other textbooks we recommend as auxiliary sources; all are freely available to UCSD personnel. Probability: Theory and Examples (5th Edition), by Rick Durrett A Probability Path by Sidney Resnick A Modern Approach to Probability Theory by Bert Fristedt and Lawrence Gray Probability Theory: A Comprehensive Course by Achim Klenke If you are not on UCSD campus, make sure you are logged into the VPN in order to gain access; you can find instructions on how to do this here .
• Coursework : There will be weekly homework assignments due on Mondays (starting in Week 2); they are posted below . There will be 5 quizzes , in weeks 1, 3, 5, 7, and 9 of the quarter; they will take place during the scheduled Thursday lecture time, with an alternate sitting available in the late evening to accommodate those in distant time-zones. And there will be a take-home final exam during exam week. Timing and due dates for all courses assessments can be found below .
• Piazza is an online discussion forum. It will allow you to post messages (openly or anonymously) and answer posts made by your fellow students, about course content, homework, quizzes, etc. The instructor and TA will also monitor and post to Piazza regularly. You can sign up here . Note: Piazza has an opt-in "Piazza Careers" section which, if you give permission, will share statistics about your Piazza use with potential future employers. It also has a "social network" component, based on other students who've shared a Piazza-based class with you, that comes with the usual warnings about privacy concerns. Piazza is fully FERPA compliant, and is an allowed resource at UCSD. Nevertheless, you are not required to use Piazza if you do not wish.
• Gradescope is an online tool for uploading and grading assignments and exams (it is now under the umbrella of Turnitin). You will turn in your homework, quizzes, and final exam through Gradescope, and you will access your graded assessments there as well. Access the class Gradescope site here .

Instructional Staff

We will be communicating with you and making announcements through an online question and answer platform called Piazza (sign up link: piazza.com/ucsd/winter2016/math180a ). We ask that when you have a question about the class that might be relevant to other students, you post your question on Piazza instead of emailing us. That way, everyone can benefit from the response. Posts about homework or exams on Piazza should be content based. While you are encouraged to crowdsource and discuss coursework through Piazza, please do not post complete solutions to homework problems there. Questions about grades should be brought to the instructors, in office hours. You can also post private messages to instructors on Piazza, which we prefer to email.

Our office hours, and all relevant scheduled course activities, can be found in the following calendar.

Recorded Lectures

The Lectures for this course are pre-recorded, and available on YouTube .

The lectures are not divided into even 80-minute chunks. They are organized by topic, concept, or example.

Below, you will find a list (with links) of the lecture videos you should watch prior to the listed date, along with pdf slides of the tablet output during those lecture videos.

Math 280A is the first quarter of a three-quarter graduate level sequence in the theory of probability. This sequence provides a rigorous treatment of probability theory, using measure theory, and is essential preparation for Mathematics PhD students planning to do research in probability. A strong background in undergraduate real analysis at the level of Math 140AB is essential for success in Math 280A. In particular, students should be comfortable with notions such as countable and uncountable sets, limsup and liminf, and open, closed, and compact sets, and should be proficient at writing rigorous epsilon-delta style proofs. Graduate students who do not have this preparation are encouraged instead to consider Math 285, a one-quarter course in stochastic processes which will be offered in Winter 2021. See also this page , maintained by Ruth Williams, for more information on graduate courses in probability at UCSD.

According to the UC San Diego Course Catalog , the topics covered in the full-year sequence 280ABC include the measure-theoretic foundations of probability theory, independence, the Law of Large Numbers, convergence in distribution, the Central Limit Theorem, conditional expectation, martingales, Markov processes, and Brownian motion. Given the current pandemic crisis and emergency remote teaching modality, it is more difficult than usual to predict what pace we will work through this material, and where the dividing line between 280A and 280B will occur.

Prerequisite:   Students should have mastered the fundamentals of real analysis in metric spaces, as covered in MATH 140AB, before taking this course. An undergraduate course in probability, comparable to MATH 180A, and further courses in stochastic processes, comparable to MATH 180BC, would also be an asset, but are not absolutely necessary.

Lectures:   The lectures for this course will be recorded asynchronously, and made available on YouTube. You should engage with the relevant videos before each "Lecture" session. The schedule Lecture times will be devoted to Q&A sessions and quizzes. The Q&A sessions will be recorded, with recordings available on Canvas; the quizzes will not be recorded (but will take place live on Zoom), and will be available in a "second sitting" to accommodate those students in far-flung time-zones.

Homework:   Homework assignments are posted below , and will be due by 9pm (with a 30-minutes "late" grace period in case of technical glitches) on Mondays throughout thee quarter. You must turn in your homework through Gradescope; if you have produced it on paper, you can scan it or simply take clear photos of it to upload. You must select pages corresponding to your solutions of problems during the upload process. Gradescope will allow you to re-select pages at any point until grading has begun. If you have not selected pages when the TA begins grading, the TA will not grade your assignment and you will receive a grade of 0 on it. No appeals of this policy will be considered. It is allowed and even encouraged to discuss homework problems with your classmates and your instructor and TA, but your final write up of your homework solutions must be your own work.

Quizzes:   There will be 5 quizzes throughout the quarter, to test your fundamental knowledge of the course material. You will write them on Thursdays 1-1:50pm or 7-7:50pm, live on Zoom (so that your instructional team can answer questions if any arise), and turn them in via Gradescope. No collaboration (with other humans or with online resources) is allowed on quizzes.

Lowest scores:    Of the 9 homework assignments, only your highest 7 scores will count towards your final grade. Of the 5 quizzes, only your highest 4 scores will count towards your final grade.

Final Exam:   The final exam will be take-home. It will be available and due during exam week; more details about the exam window will be available later in the term. No collaboration (with other humans or with online resources) is allowed on the final exam. We reserve the right to invite students to follow-up Zoom meetings after the final exam to confirm that the work was completed without collaboration. We reserve the scheduled final exam time-slot for this purpose.

• Exam Responsibilities   An outline of the responsibilities of faculty and students with regard to final exams
• Policies on Examinations   The Academic Senate policy regarding final examinations (These are the rules!)

Regrade Policy:    Your quizzes, homeworks, and final exam will be graded using Gradescope . For quizzes and the final exam , you will be able to request regrades through Gradescope for a specified window of time. Be sure to make your request within the specified window of time; no regrade requests will be accepted after the deadline. For homework , any clerical erros (such as a problem or page that the TA accidentally missed when grading) should be discussed with the TA during office hours. Grading rubrics are not negotiable ; if the TA has taken off some number of points from your solution, there is a sound pegagogical reason for this. This is a PhD class in mathematics . We are not focused on numerical grades here; we are focused on learning deep and challenging material. The grading is meant as a formative assessment tool; if your grade is not perfect, it indicates you should spend more time reviewing the concepts and thinking about the problems. The TA will give detailed feedback in the grading; it is your responsibility to think and work hard to understand what concepts and ideas you need a firmer understanding of from any assignment where you did not receive full points. Only after working hard on your own, or in collaboration with fellow classmates (for example through Piazza), should you consider approaching your TA or instructor for further explanation of grading choices. However, please understand that these conversations will not result in a change in your grade unless there has been some clear clerical error, such as the TA accidentally missing part of your solution. The TA will not change their assessment of a students work due to conversations or complaints after the fact.

• 40% Homework,  30% Quizzes,  30% Final Exam

Academic Integrity:   UC San Diego's code of academic integrity outlines the expected academic honesty of all students and faculty, and details the consequences for academic dishonesty. The main issues are cheating and plagiarism, of course, for which we have a zero-tolerance policy. (Penalties for these offenses always include assignment of a failing grade in the course, and usually involve an administrative penalty, such as suspension or expulsion, as well.) However, academic integrity also includes things like giving credit where credit is due (listing your collaborators on homework assignments, noting books or papers containing information you used in solutions, etc.), and treating your peers and instructors respectfully in all forms of interaction (Zoom meetings, email, Piazza discussions, etc).

Assessment Versioning : following UCSD (and common) practice, recommended by the Academic Integrity Office, assessments given at non-overlapping times will be comparable , but may not be identical . In particular: the two sittings of each Quiz (and potentially quizzes given within each sitting) may not have exactly the same questions, but will be designed to cover the same material and be of equivalent levels of difficulty. This practice is meant to maintain course integrity, avoiding unpermitted collaboration (either intentional or accidental).

OSD Accommodations: Students requesting accommodations for this course due to a disability must provide a current Authorization for Accommodation (AFA) letter issued by the Office for Students with Disabilities (OSD) which is located in University Center 202 behind Center Hall. The AFA letter will be issued by the OSD electronically. Please make arrangements to discuss your accommodations with me in advance ( by the end of Week 2 ). We will make every effort to arrange for whatever accommodations are stipulated by your AFA letter. For more information, see here .

Covid-19 Accommodations: Due to the unprecedented Covid-19 pandemic, our lectures and all meetings will be remote, using Zoom, this year. Some of you are residing in different time-zones (and different continents), and these present additional challenges. To accommodate, all lectures are pre-recorded asynchronously, and the Q&A sessions are also recorded. Office hours will not be recorded, but will be distributed throughout different days and times so all students should be able to attend in regular work hours; failing that, individual Zoom meetings can be made. Synchronous quizzes will be held in two "sittings": 1-1:50pm and 7-7:50pm Pacific Time, which should cover everyone enrolled in the class, during daylight hours. If these accommodations are still insufficient due to a severe Covid-19 pandemic related issue you have, please contact me no later than Friday, October 16 , to discuss other arrangements.

Weekly homework assignments are posted here. Homework is due by 9:00pm on the posted date, through Gradescope . Late homework will not be accepted.

• Homework 1 , due Monday, October 12.
• Homework 2 , due Monday, October 19.
• Homework 3 , due Monday, October 26.
• Homework 4 , due Monday, November 2.
• Homework 5 , due Monday, November 9.
• Homework 6 , due Monday, November 16.
• Homework 7 , due Monday, November 23.
• Homework 8 , due Monday, November 30.
• Homework 9 , due Monday, December 7.
• Department of Mathematics, UC San Diego

Free probability simulations for 7th grade

These interactive probability simulations work in MS Excel, LibreOffice Calc, or any other spreadsheet program that can open xls files. Feel free to download them and use them with your students.

1. Dice roller

This is a simple virtual die roller (one die). It includes different sheets for 500 rolls, 1000 rolls, and 2000 rolls, so you can compare the three, and see how the experimental probabilities gets closer to the theoretical ones as we increase the number of rolls. The spreadsheet automatically calculates the frequencies for all six outcomes (1, 2, 3, 4, 5, 6) and their experimental probabilities.

Click to download or open Dice-roller.xls (Excel file).

2. Two-coin toss

This simulation uses 1s and 0s to represent heads and tails of two coins. It contains two sheets: one for 200 tosses and the other for 500 tosses. The spreadsheet automatically calculates the frequencies for all four outcomes (HH, HT, TH, TT) and their experimental probabilities.

Click to download or open two-coin-toss.xls (Excel file).

3. Number of males and females in a sample of 10 people

If you choose 10 people randomly, what is the probability that exactly 5 of them are male and 5 are female? Or that exactly 7 or them are male and 3 female? This simulation explores the situation with either 100, 500, or 2000 repetitions (or samples).

Click to download or open 10-people-females-males.xls (Excel file).

4. Sample of six students

This is a more complex simulation. We choose a sample of six students from a student population where the probability that a single student completed homework on time is 50%. The sampling is repeated either 100 or 500 times. We observe how many students out of the six completed homework on time.

The simulation uses random digits so that 0 represents a student who didn't complete homework on time and 1 represents a student who did. The spreadsheet automatically calculates the frequencies of each outcome (0-6 students), the probabilities, and draws a bar graph for the distribution.

The second sheet in the spreadsheet explores the same situation, but this time the probability that a single student completed homework on time is 70%.

Click to download or open random-digits-six-students.xls (Excel file).

Probability video lessons by Maria

By Maria Miller

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Discrete probability distributions

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Continuous probability distributions

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Special-purpose calculators

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Each calculator features clear instructions, answers to frequently-asked questions, and a one or more problems with solutions to illustrate calculator use.

Probability Calculator

Probability of two events.

To find out the union, intersection, and other related probabilities of two independent events.

Probability Solver for Two Events

Please provide any 2 values below to calculate the rest probabilities of two independent events.

Probability of a Series of Independent Events

Probability of a normal distribution.

Use the calculator below to find the area P shown in the normal distribution, as well as the confidence intervals for a range of confidence levels.

Related Standard Deviation Calculator | Sample Size Calculator | Statistics Calculator

Probability is the measure of the likelihood of an event occurring. It is quantified as a number between 0 and 1, with 1 signifying certainty, and 0 signifying that the event cannot occur. It follows that the higher the probability of an event, the more certain it is that the event will occur. In its most general case, probability can be defined numerically as the number of desired outcomes divided by the total number of outcomes. This is further affected by whether the events being studied are independent, mutually exclusive, or conditional, among other things. The calculator provided computes the probability that an event A or B does not occur, the probability A and/or B occur when they are not mutually exclusive, the probability that both event A and B occur, and the probability that either event A or event B occurs, but not both.

Complement of A and B

Given a probability A , denoted by P(A) , it is simple to calculate the complement, or the probability that the event described by P(A) does not occur, P(A') . If, for example, P(A) = 0.65 represents the probability that Bob does not do his homework, his teacher Sally can predict the probability that Bob does his homework as follows:

P(A') = 1 - P(A) = 1 - 0.65 = 0.35

Given this scenario, there is, therefore, a 35% chance that Bob does his homework. Any P(B') would be calculated in the same manner, and it is worth noting that in the calculator above, can be independent; i.e. if P(A) = 0.65, P(B) does not necessarily have to equal 0.35 , and can equal 0.30 or some other number.

Intersection of A and B

The intersection of events A and B , written as P(A ∩ B) or P(A AND B) is the joint probability of at least two events, shown below in a Venn diagram. In the case where A and B are mutually exclusive events, P(A ∩ B) = 0 . Consider the probability of rolling a 4 and 6 on a single roll of a die; it is not possible. These events would therefore be considered mutually exclusive. Computing P(A ∩ B) is simple if the events are independent. In this case, the probabilities of events A and B are multiplied. To find the probability that two separate rolls of a die result in 6 each time:

The calculator provided considers the case where the probabilities are independent. Calculating the probability is slightly more involved when the events are dependent, and involves an understanding of conditional probability, or the probability of event A given that event B has occurred, P(A|B) . Take the example of a bag of 10 marbles, 7 of which are black, and 3 of which are blue. Calculate the probability of drawing a black marble if a blue marble has been withdrawn without replacement (the blue marble is removed from the bag, reducing the total number of marbles in the bag):

Probability of drawing a blue marble:

P(A) = 3/10

Probability of drawing a black marble:

P(B) = 7/10

Probability of drawing a black marble given that a blue marble was drawn:

P(B|A) = 7/9

As can be seen, the probability that a black marble is drawn is affected by any previous event where a black or blue marble was drawn without replacement. Thus, if a person wanted to determine the probability of withdrawing a blue and then black marble from the bag:

Probability of drawing a blue and then black marble using the probabilities calculated above:

P(A ∩ B) = P(A) × P(B|A) = (3/10) × (7/9) = 0.2333

Union of A and B

In probability, the union of events, P(A U B) , essentially involves the condition where any or all of the events being considered occur, shown in the Venn diagram below. Note that P(A U B) can also be written as P(A OR B) . In this case, the "inclusive OR" is being used. This means that while at least one of the conditions within the union must hold true, all conditions can be simultaneously true. There are two cases for the union of events; the events are either mutually exclusive, or the events are not mutually exclusive. In the case where the events are mutually exclusive, the calculation of the probability is simpler:

A basic example of mutually exclusive events would be the rolling of a dice, where event A is the probability that an even number is rolled, and event B is the probability that an odd number is rolled. It is clear in this case that the events are mutually exclusive since a number cannot be both even and odd, so P(A U B) would be 3/6 + 3/6 = 1 , since a standard dice only has odd and even numbers.

The calculator above computes the other case, where the events A and B are not mutually exclusive. In this case:

Using the example of rolling dice again, find the probability that an even number or a number that is a multiple of 3 is rolled. Here the set is represented by the 6 values of the dice, written as:

Exclusive OR of A and B

Another possible scenario that the calculator above computes is P(A XOR B) , shown in the Venn diagram below. The "Exclusive OR" operation is defined as the event that A or B occurs, but not simultaneously. The equation is as follows:

As an example, imagine it is Halloween, and two buckets of candy are set outside the house, one containing Snickers, and the other containing Reese's. Multiple flashing neon signs are placed around the buckets of candy insisting that each trick-or-treater only takes one Snickers OR Reese's but not both! It is unlikely, however, that every child adheres to the flashing neon signs. Given a probability of Reese's being chosen as P(A) = 0.65 , or Snickers being chosen with P(B) = 0.349 , and a P(unlikely) = 0.001 that a child exercises restraint while considering the detriments of a potential future cavity, calculate the probability that Snickers or Reese's is chosen, but not both:

0.65 + 0.349 - 2 × 0.65 × 0.349 = 0.999 - 0.4537 = 0.5453

Therefore, there is a 54.53% chance that Snickers or Reese's is chosen, but not both.

Normal Distribution

The normal distribution or Gaussian distribution is a continuous probability distribution that follows the function of:

where μ is the mean and σ 2 is the variance. Note that standard deviation is typically denoted as σ . Also, in the special case where μ = 0 and σ = 1 , the distribution is referred to as a standard normal distribution. Above, along with the calculator, is a diagram of a typical normal distribution curve.

The normal distribution is often used to describe and approximate any variable that tends to cluster around the mean, for example, the heights of male students in a college, the leaf sizes on a tree, the scores of a test, etc. Use the "Normal Distribution" calculator above to determine the probability of an event with a normal distribution lying between two given values (i.e. P in the diagram above); for example, the probability of the height of a male student is between 5 and 6 feet in a college. Finding P as shown in the above diagram involves standardizing the two desired values to a z-score by subtracting the given mean and dividing by the standard deviation, as well as using a Z-table to find probabilities for Z. If, for example, it is desired to find the probability that a student at a university has a height between 60 inches and 72 inches tall given a mean of 68 inches tall with a standard deviation of 4 inches, 60 and 72 inches would be standardized as such:

Given μ = 68; σ = 4 (60 - 68)/4 = -8/4 = -2 (72 - 68)/4 = 4/4 = 1

The graph above illustrates the area of interest in the normal distribution. In order to determine the probability represented by the shaded area of the graph, use the standard normal Z-table provided at the bottom of the page. Note that there are different types of standard normal Z-tables. The table below provides the probability that a statistic is between 0 and Z, where 0 is the mean in the standard normal distribution. There are also Z-tables that provide the probabilities left or right of Z, both of which can be used to calculate the desired probability by subtracting the relevant values.

For this example, to determine the probability of a value between 0 and 2, find 2 in the first column of the table, since this table by definition provides probabilities between the mean (which is 0 in the standard normal distribution) and the number of choices, in this case, 2. Note that since the value in question is 2.0, the table is read by lining up the 2 row with the 0 column, and reading the value therein. If, instead, the value in question were 2.11, the 2.1 row would be matched with the 0.01 column and the value would be 0.48257. Also, note that even though the actual value of interest is -2 on the graph, the table only provides positive values. Since the normal distribution is symmetrical, only the displacement is important, and a displacement of 0 to -2 or 0 to 2 is the same, and will have the same area under the curve. Thus, the probability of a value falling between 0 and 2 is 0.47725 , while a value between 0 and 1 has a probability of 0.34134. Since the desired area is between -2 and 1, the probabilities are added to yield 0.81859, or approximately 81.859%. Returning to the example, this means that there is an 81.859% chance in this case that a male student at the given university has a height between 60 and 72 inches.

The calculator also provides a table of confidence intervals for various confidence levels. Refer to the Sample Size Calculator for Proportions for a more detailed explanation of confidence intervals and levels. Briefly, a confidence interval is a way of estimating a population parameter that provides an interval of the parameter rather than a single value. A confidence interval is always qualified by a confidence level, usually expressed as a percentage such as 95%. It is an indicator of the reliability of the estimate.

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