If n independent trials are conducted of an experiment for which the success probability is p on each trial, then the expected number of successes to occur during the n trials is given by
E = np.
Example 5.42. Remember the basketball player named Karen who is an 80% free throw shooter and who attempted six shots in a game? (See Example 5.36.) What would be the expected value for the number of shots she hits in the six attempts?
Solution: Here we have an independent trials process with n = 6 and p = .8. So the expected number of shots that she would hit would be
E = np = 6 × .8 = 4.8.
Problems
1. Edith is throwing darts at a target. On each throw she has a 40% chance of hitting the target.
(a) What is the probability she hits exactly 2 of the 5 throws? (b) In 5 throws, what is the expected number of times she will hit?
2. Mark is a 70% free throw shooter in basketball. If he goes to the foul line to shoot a “one-and-one”,
Section 5.7: Expected Value 207 (b) what is the expected number of shots he will take?
[Note: A one-and-one means that he shoots once and if he hits he gets another shot.] 3. Three male and three female patients are seated in a doctor’s waiting room. Patients are randomly called in to see the doctor. As soon as she has treated two of the female patients, the doctor plans to take a coffee break.
(a) Draw a tree diagram to represent the possibilities for the patients treated before the doctor’s coffee break.
(b) What is the expected number of patients the doctor will treat before her coffee break?
(c) What is the expected number of male patients the doctor will treat before her coffee break? [Can you see the reason for the relationship between this answer and the answer to part (b)?]
4. Four eggs are left in a carton. Two of them are rotten. You start cracking the eggs one at a time until you have cracked both rotten eggs or both good eggs.
(a) Draw a tree diagram to represent this process. (b) What is the probability you have to crack 3 eggs?
(c) What is the expected value for the number of eggs you will crack?
(d) What is the expected value for the number of rotten eggs that will be cracked? 5. Each day that Sue drives to NCSU, there is a 60% chance that she will have to stop
at the Western Boulevard stoplight. If she drives to the university 5 days in a given week,
(a) what is the probability she will get caught by the light exactly 4 times? (b) what is the probability she will get caught by the light at least 4 times? (c) what is the expected value for the number of times she will have to stop?
6. Sam buys a raffle ticket for $1. 1000 of the tickets are to be sold. First prize is a TV valued at $350, second prize is a tape deck worth $115, and third prize is a camera worth $40.
(a) What is the expected value of the raffle ticket?
(b) If Sam buys 5 tickets, what is the probability he wins a prize?
7. The number of traffic accidents in Fuquay-Varina during rush hour on Friday after- noons is 0, 1, 2, or 3 with probabilities .94, .03, .02, and .01 respectively.
(a) Find the expected number of accidents during the Friday afternoon rush hour. (b) How many accidents on Friday afternoons would you expect during a year?
(Hint: How many Friday afternoons are there during a year?)
8. Bill’s dresser drawer contains 7 blue socks and 3 white ones. He reaches into the drawer and pulls out socks one at a time until he has two that match.
(a) Draw a tree diagram for this process.
(b) What is the probability he gets a pair of white socks?
(c) What is the probability he will get a match with the first two socks? (d) What is the expected number of socks he will pull from the drawer?
9. A student is trying to pass a competency exam in French. Each time she takes the exam she has a 25% chance of passing, and she is allowed a maximum of three attempts.
(a) Draw a tree diagram to represent her attempts to pass the exam. (b) What is the probability she will eventually pass the exam? (c) What is the probability she will take the exam three times? (d) What is the expected number of times she will take the exam? (e) What is the expected number of times she will fail the exam?
10. If 2 cards are dealt from a standard deck of 52 cards, what is the expected number of spades dealt?
11. 3 men and 4 women are at a small party at which 3 door prizes will be awarded. Nobody will receive more than one prize.
(a) Find the probability that none of the prizes is won by a man.
(b) Find the probability that exactly one of the prizes is won by a man. (c) Find the probability that exactly two of the prizes are won by men. (d) Find the probability that all three prizes are won by men.
(e) What is the expected value for the number of prizes to be won by men? 12. In Problem 13 of Section 5.4,
(a) what is the expected value for the number of mints chosen?
(b) what is the expected value for the number of candies chosen that are wrapped in red foil?
13. Paul has three $5 bills and three $10 bills in his pocket. Nick has four $1 bills and two $20 bills in his pocket. Each person pulls out a bill from his own pocket. What is the expected value for the amount of money that Paul will receive in each of the following situations?
(a) The person that pulls out the lower amount pays the other person $5.
(b) The person that pulls out the lower amount pays the other person a number of dollars equal to the difference between the two amounts.
(c) The person that pulls out the lower amount pays the other person the amount of money shown on the bill pulled out by the other person.
14. Tanya wants to pass a qualifying exam, and she is allowed 3 attempts at passing the exam if necessary. Each time she takes the exam there is a 40% chance she will pass. What is the expected number of times she will take the exam? [Hint: Draw a tree.]
Section 5.7: Expected Value 209
Chapter 5 Review Problems
1. John, Ed, Ann, and Barbara are lining up in a random order to buy tickets to a movie. What is the probability that they will line up in alphabetical order?
2. John and Jerry are 2 applicants among 40 applicants at a company that plans to hire 10 people. If the people chosen are randomly selected from among the 40 applicants,
(a) what is the probability that John will get a job?
(b) what is the probability that both John and Jerry will get jobs? (c) what is the probability that neither will get a job?
3. The board of directors authorizes the bank where Sam works to promote 3 of their 10 junior executives to the rank of vice-president. The promotions will be random. (a) What is the probability that Sam will be promoted?
(b) What is the probability that Sam won’t be promoted but that his arch rival Carl will be?
4. To play a particular card game, a player is dealt 2 cards from an ordinary 52-card deck.
(a) Disregarding the order in which the cards are dealt, how many different hands are possible?
(b) What is the probability that a given hand (of 2 cards) contains 2 spades? (c) What is the probability that the player will be dealt two cards of the same suit? 5. There are 4 seasons (spring, summer, winter, fall). Assume that people are equally
likely to be born in each of the seasons. Find the probability that in a group of 3 people, at least 2 were born in the same season.
6. Four salesmen play “odd man out” to see who pays for lunch. They each flip a coin, and if there is a salesman who doesn’t match the others he pays for lunch. (For instance, he might get “heads” while the other three get “tails.”) What is the probability that there is an “odd man” the first time they flip?
7. An ordinary die is rolled twice.
(a) What is the probability that the sum of the two numbers obtained is equal to 9? (b) What is the probability the sum is 9 if you know that the first roll showed a 5? (c) What is the probability the sum is 9 if you know that the second roll showed a
5?
(d) What is the probability the sum is 9 if you know that at least one of the rolls showed a 5?
(e) What is the probability the sum is 9 if you know that the first roll showed a 2? (f) What is the expected value for the sum?
8. Of a group of students applying to a particular college, 55% were female and 45% were male. Of the female applicants 40% were accepted, and of the male applicants 35% were accepted.
(a) What percentage of applicants were accepted?
(b) If someone tells you that her cousin applied and was accepted, what is the probability that her cousin is female?
9. A submarine is firing torpedoes at a ship. Each has probability .6 of hitting the ship. How many must be fired in order to be 95% certain of scoring a hit?
10. A basketball player hits 70% of his free throws. If his attempts are considered independent trials and he makes 5 attempts in a game,
(a) what is the probability he hits exactly 4 shots? (b) what is the probability he hits at least 4 shots?
(c) what is the expected value for the number of shots he hits?
11. In a box containing 5 pieces of chocolate candy that are all wrapped alike, Kay knows that there are 2 pieces that she likes. She is determined to keep opening pieces of candy until she finds these 2 pieces.
(a) Draw a tree for this process of opening pieces of candy.
(b) What is the probability that she opens exactly 4 pieces of candy?
(c) What is the probability that she opens exactly 4 pieces if we know that she didn’t like the first piece she opened?
(d) If she opened exactly 3 pieces to find the 2 that she liked, what is the probability that she liked the first piece she opened?
(e) What is the expected number of pieces for her to open?
12. Two cards are drawn without replacement from a deck of cards. What is the probability
(a) both cards are face cards (kings, queens, and jacks)?
(b) both cards are face cards given that at least one is a face card? (c) both are face cards given that at least one of the cards is a king?
13. Fred and Jerry like to go to trap shoots together. There are 3 events at a trap shoot, the 16-yard event, handicap, and doubles. Fred beats Jerry half the time in the 16- yard event, 2/3 of the time in the handicap event, and 3/5 of the time in the doubles event. If the results in the 3 events are independent, what is the probability that at a particular trap shoot
(a) Fred beats Jerry in all three events? (b) Fred beats Jerry in at least two events?
(c) Fred beats Jerry in at least two events given that Fred beats Jerry in the 16-yard event?
14. Susan has 2 black pens and 3 red pens in her purse. She pulls out pens one at a time without replacement until she has a pen of each color.
Chapter 5 Review Problems 211 (a) Draw the probability tree diagram.
(b) What is the probability she pulls out at least 3 pens?
(c) What is the probability she pulls out at least 3 pens given the first pen she chooses is red?
(d) What is the probability the first pen she chooses is red given that she pulls out at least 3 pens?
(e) Let A be the event “the first pen chosen is red” and let B be the event “at least 3 pens are chosen”. Are A and B independent? Why or why not?
(f) What is the expected number of pens she chooses? (g) What is the expected number of red pens she chooses?
15. You have to answer 5 multiple-choice questions. Each question has 4 different answers, only one of which is correct. If you answer them by randomly guessing, (a) what is the probability you will get all 5 correct?
(c) what is the probability you will get at least 4 correct?