Random Number Generator A random number generator on a computer selects two integers

An ________ is any happening for which the result is uncertain, and the possible results are called ________.

The set of all possible outcomes of an experiment is called the ________ ________.

To determine the ________ of an event, you can use the formula where is the number of outcomes in the event and is the number of outcomes in the sample space.

If then is an ________ event, and if then is a ________ event.

If two events from the same sample space have no outcomes in common, then the two events are ________ ________.

If the occurrence of one event has no effect on the occurrence of a second event, then the events are ________.

The ________ of an event is the collection of all outcomes in the sample space that are not in A.

Match the probability formula with the correct probability name. (a) Probability of the union of two events (i) (b) Probability of mutually exclusive events (ii) (c) Probability of independent events (iii) (d) Probability of a complement

You toss a coin and a six-sided die.

You toss a six-sided die twice and record the sum of the results.

A taste tester ranks three varieties of yogurt, A, B, and C, according to preference.

You select two marbles (without replacement) from a bag containing two red marbles, two blue marbles, and one yellow marble. You record the color of each marble.

Two county supervisors are selected from five supervisors, A, B, C, D, and E, to study a recycling plan.

A sales representative makes presentations about a product in three homes per day. In each home, there may be a sale (denote by S) or there may be no sale (denote by F).

The probability of getting exactly one tail

The probability of getting exactly two tails

The probability of getting a head on the first toss

The probability of getting a tail on the last toss

The probability of getting at least one head

The probability of getting at least two heads

The card is a face card.

The card is not a face card.

The card is a red face card.

The card is a 9 or lower. (Aces are low.)

The sum is 6. 2

The sum is at least 8.

The sum is less than 11.

The sum is 2, 3, or 12.

The sum is odd and no more than 7.

The sum is odd or prime.

Both marbles are red. 3

Both marbles are yellow.

Neither marble is yellow.

The marbles are different colors.

Graphical Reasoning In 2011, there were approximately 13.75 million unemployed workers in the United States. The circle graph shows the age profile of these unemployed workers. (Source: U.S. Bureau of Labor Statistics) (a) Estimate the number of unemployed workers in the 1619 age group. (b) What is the probability that a person selected at random from the population of unemployed workers is in the 2544 age group? (c) What is the probability that a person selected at random from the population of unemployed workers is in the 4564 age group? (d) What is the probability that a person selected at random from the population of unemployed workers is 45 or older?

Data Analysis An independent polling organization interviews one hundred college students to determine their political party affiliations and whether they favor a balanced-budget amendment to the Constitution. The table lists the results of the study. In the table, represents Democrat and represents Republican. Find the probability that a person selected at random from the sample is as described. (a) A person who does not favor the amendment (b) A Republican (c) A Democrat who favors the amendment

Education In a high school graduating class of 128 students, 52 are on the honor roll. Of these, 48 are going on to college; of the other 76 students, 56 are going on to college. What is the probability that a student selected at random from the class is (a) going to college, (b) not going to college, and (c) not going to college and on the honor roll?

Alumni Association A college sends a survey to members of the class of 2012. Of the 1254 people who graduated that year, 672 are women, of whom 124 went on to graduate school. Of the 582 male graduates, 198 went on to graduate school. What is the probability that a class of 2012 alumnus selected at random is (a) female, (b) male, and (c) female and did not attend graduate school?

Winning an Election Three people are running for president of a class. The results of a poll indicate that the first candidate has an estimated 37% chance of winning and the second candidate has an estimated 44% chance of winning. What is the probability that the third candidate will win?

Payroll Error The employees of a company work in six departments: 31 are in sales, 54 are in research, 42 are in marketing, 20 are in engineering, 47 are in finance, and 58 are in production. The payroll department loses one employees paycheck. What is the probability that the employee works in the research department?

A class receives a list of 20 study problems, from which 10 will be part of an upcoming exam. A student knows how to solve 15 of the problems. Find the probability that the student will be able to answer (a) all 10 questions on the exam, (b) exactly eight questions on the exam, and (c) at least nine questions on the exam.

A payroll department addresses five paychecks and envelopes to five different people and randomly inserts the paychecks into the envelopes. What is the probability that (a) exactly one paycheck is inserted in the correct envelope and (b) at least one paycheck is in the correct envelope?

On a game show, you are given five digits to arrange in the proper order to form the price of a car. If you are correct, then you win the car. What is the probability of winning, given the following conditions? (a) You guess the position of each digit. (b) You know the first digit and guess the positions of the other digits.

The deck for a card game is made up of 108 cards. Twenty-five each are red, yellow, blue, and green, and eight are wild cards. Each player is randomly dealt a seven-card hand. (a) What is the probability that a hand will contain exactly two wild cards? (b) What is the probability that a hand will contain two wild cards, two red cards, and three blue cards?

You draw one card at random from a standard deck of 52 playing cards. Find the probability that (a) the card is an even-numbered card, (b) the card is a heart or a diamond, and (c) the card is a nine or a face card.

You draw five cards at random from a standard deck of 52 playing cards. What is the probability that the hand drawn is a full house? (A full house is a hand that consists of two of one kind and three of another kind.)

A shipment of 12 microwave ovens contains three defective units. A vending company has ordered four units, and because each has identical packaging, the selection will be random. What is the probability that (a) all four units are good, (b) exactly two units are good, and (c) at least two units are good?

ATM personal identification number (PIN) codes typically consist of four-digit sequences of numbers. Find the probability that if you forget your PIN, then you can guess the correct sequence (a) at random and (b) when you recall the first two digits.

Random Number Generator A random number generator on a computer selects two integers from 1 through 40. What is the probability that (a) both numbers are even, (b) one number is even and one number is odd, (c) both numbers are less than 30, and (d) the same number is selected twice?

Flexible Work Hours In a survey, people were asked whether they would prefer to work flexible hourseven when it meant slower career advancementso they could spend more time with their families. The figure shows the results of the survey. What is the probability that three people chosen at random would prefer flexible work hours?

PE 0.87 P

PE 0.36

P 3 E 1 4

PE 2 P 3

PE 0.23 P

PE 0.92 P

P 100 E 17 35

PE 61 P 100

Backup System A space vehicle has an independent backup system for one of its communication networks. The probability that either system will function satisfactorily during a flight is 0.985. What is the probability that during a given flight (a) both systems function satisfactorily, (b) at least one system functions satisfactorily, and (c) both systems fail?

Backup Vehicle A fire company keeps two rescue vehicles. Because of the demand on the vehicles and the chance of mechanical failure, the probability that a specific vehicle is available when needed is 90%. The availability of one vehicle is independent of the availability of the other. Find the probability that (a) both vehicles are available at a given time, (b) neither vehicle is available at a given time, and (c) at least one vehicle is available at a given time.

Roulette American roulette is a game in which a wheel turns on a spindle and is divided into 38 pockets. Thirty-six of the pockets are numbered 136, of which half are red and half are black. Two of the pockets are green and are numbered 0 and 00 (see figure). The dealer spins the wheel and a small ball in opposite directions. As the ball slows to a stop, it has an equal probability of landing in any of the numbered pockets. (a) Find the probability of landing in the number 00 pocket. (b) Find the probability of landing in a red pocket. (c) Find the probability of landing in a green pocket or a black pocket. (d) Find the probability of landing in the number 14 pocket on two consecutive spins. (e) Find the probability of landing in a red pocket on three consecutive spins.

A Boy or a Girl? Assume that the probability of the birth of a child of a particular sex is 50%. In a family with four children, what is the probability that (a) all the children are boys, (b) all the children are the same sex, and (c) there is at least one boy?

Geometry You and a friend agree to meet at your favorite fast-food restaurant between 5:00 P.M. and 6:00 P.M. The one who arrives first will wait 15 minutes for the other, and then will leave (see figure). What is the probability that the two of you will actually meet, assuming that your arrival times are random within the hour?

Estimating You drop a coin of diameter onto a paper that contains a grid of squares units on a side (see figure). (a) Find the probability that the coin covers a vertex of one of the squares on the grid. (b) Perform the experiment 100 times and use the results to approximate

If and are independent events with nonzero probabilities, then can occur when occurs.

Rolling a number less than 3 on a normal six-sided die has a probability of . The complement of this event is to roll a number greater than 3, and its probability

Pattern Recognition Consider a group of people. (a) Explain why the following pattern gives the probabilities that the people have distinct birthdays. (b) Use the pattern in part (a) to write an expression for the probability that people have distinct birthdays. (c) Let be the probability that the people have distinct birthdays. Verify that this probability can be obtained recursively by and (d) Explain why gives the probability that at least two people in a group of people have the same birthday. (e) Use the results of parts (c) and (d) to complete the table. (f) How many people must be in a group so that the probability of at least two of them having the same birthday is greater than Explain.

HOW DO YOU SEE IT? The circle graphs show the percents of undergraduate students by class level at two colleges. A student is chosen at random from the combined undergraduate population of the two colleges. The probability that the student is a freshman, sophomore, or junior is 81%. Which college has a greater number of undergraduate students? Explain your reasoning.