Chapter 4

Two balls are chosen randomly from an urn containing 8 white, 4 black, and 2 orange balls. Suppose that we win \(\$ 2\) for each black ball selected and we lose \(\$ 1\) for each white ball selected. Let \(X\) denote our winnings. What are the possible values of \(X\), and what are the probabilities associated with each value?

Two fair dice are rolled. Let \(X\) equal the product of the 2 dice. Compute \(P\\{X=i\\}\) for \(i=1, \ldots, 36\)

Five men and 5 women are ranked according to their scores on an examination. Assume that no two scores are alike and all \(10 !\) possible rankings are equally likely. Let \(X\) denote the highest ranking achieved by a woman. (For instance, \(X=1\) if the top-ranked person is female.) Find \(\mathrm{P}\\{\mathrm{X}=\mathrm{i}\\}\) \(i=1,2,3, \ldots, 8,9,10\)

Let \(X\) represent the difference between the number of heads and the number of tails obtained when a coin is tossed \(n\) times. What are the possible values of \(X ?\)

In Problem \(5,\) for \(n=3,\) if the coin is assumed fair, what are the probabilities associated with the values that \(X\) can take on?

Suppose that a die is rolled twice. What are the possible values that the following random variables can take on: (a) the maximum value to appear in the two rolls; (b) the minimum value to appear in the two rolls; (c) the sum of the two rolls; (d) the value of the first roll minus the value of the second roll?

In the game of Two-Finger Morra, 2 players show 1 or 2 fingers and simultancously guess the number of fingers their opponent will show. If only one of the players guesses correctly, he wins an amount (in dollars) equal to the sum of the fingers shown by him and his opponent. If both players guess correctly or if neither guesses correctly, then no money is exchanged. Consider a specificd player, and denote by \(X\) the amount of money he wins in a single game of Two-Finger Morra. (a) If each player acts independently of the other, and if each player makes his choice of the number of fingers he will hold up and the number he will guess that his opponent will hold up in such a way that each of the 4 possibilities is equally likely, what are the possible values of \(X\) and what are their associated probabilities? (b) Suppose that each player acts independently of the other. If each player decides to hold up the same number of fingers that he guesses his opponent will hold up, and if each player is equally likely to hold up 1 or 2 fingers, what are the possible values of \(X\) and their associated probabilities?

A salesman has scheduled two appointments to sell encyclopedias. His first appointment will lead to a sale with probability \(.3,\) and his second will lead independently to a sale with probability .6. Any sale made is equally likely to be either for the deluxe model, which costs \(\$ 1000\), or the standard model, which costs \(\$ 500 .\) Determine the probability mass function of \(X,\) the total dollar value of all sales.

Five distinct numbers are randomly distributed to players numbered 1 through \(5 .\) Whenever two players compare their numbers, the one with the higher one is declared the winner. Initially, players 1 and 2 compare their numbers; the winner then compares her number with that of player \(3,\) and so on. Let \(X\) denote the number of times player 1 is a winner. Find \(P\\{X=i\\}, i=0,1,2,3,4\)

The National Basketball Association (NBA) draft lottery involves the 11 teams that had the worst won-lost records during the year. A total of 66 balls are placed in an urn. Each of these balls is inscribed with the name of a team: Eleven have the name of the team with the worst record, 10 have the name of the team with the second-worst record, 9 have the name of the team with the thirdworst record, and so on (with 1 ball having the name of the team with the 11 th-worst record). A ball is then chosen at random, and the team whose name is on the ball is given the first pick in the draft of players about to enter the league. Another ball is then chosen, and if it "belongs" to a team different from the one that received the first draft pick, then the team to which it belongs receives the second draft pick. (If the ball belongs to the team receiving the first pick, then it is discarded and another one is chosen; this continues until the ball of another team is chosen.) Finally, another ball is chosen, and the team named on the ball (provided that it is different from the previous two teams) receives the third draft pick. The remaining draft picks 4 through 11 are then awarded to the 8 teams that did not "win the lottery," in inverse order of their won-lost records. For instance, if the team with the worst record did not receive any of the 3 lottery picks, then that team would receive the fourth draft pick. Let \(X\) denote the draft pick of the team with the worst record. Find the probability mass function of \(X\)

Suppose that the distribution function of \(X\) is given by $$F(b)=\left\\{\begin{array}{ll} 0 & b<0 \\ \frac{b}{4} & 0 \leq b<1 \\ \frac{1}{2}+\frac{b-1}{4} & 1 \leq b<2 \\ \frac{11}{12} & 2 \leq b<3 \\ 1 & 3 \leq b \end{array}\right.$$ (a) Find \(P\\{X=i\\}, i=1,2,3\) (b) Find \(P\left\\{\frac{1}{2}

Four independent flips of a fair coin are made. Let \(X\) denote the number of heads obtained. Plot the probability mass function of the random variable \(x-2\)

A gambling book recommends the following "winning strategy" for the game of roulette: Bet \(\$ 1\) on red. If red appears (which has probability \(\frac{18}{38}\) ), then take the \(\$ 1\) profit and quit. If red does not appear and you lose this bet (which has probability \(\frac{20}{38}\) of occurring , make additional \(\$ 1\) bets on red on each of the next two spins of the roulette wheel and then quit. Let \(X\) denote your winnings when you quit. (a) Find \(P\\{X>0\\}\) (b) Are you convinced that the strategy is indeed a "winning" strategy? Explain your answer! (c) Find \(E[X]\)

Four buses carrying 148 students from the same school arrive at a football stadium. The buses carry, respectively, \(40,33,25,\) and 50 students. One of the students is randomly selected. Let \(X\) denote the number of students that were on the bus carrying the randomly selected student. One of the 4 bus drivers is also randomly selected. Let \(Y\) denote the number of students on her bus. (a) Which of \(E[X]\) or \(E[Y]\) do you think is larger? Why? (b) Compute \(E[X]\) and \(E[Y]\)

Suppose that two teams play a series of games that ends when one of them has won \(i\) games. Suppose that each game played is, independently, won by team \(A\) with probability \(p .\) Find the expected number of games that are played when (a) \(i=2\) and (b) \(i=3 .\) Also, show in both cases that this number is maximized when \(p=\frac{1}{2}\)

You have \(\$ 1000,\) and a certain commodity presently sells for \(\$ 2\) per ounce. Suppose that after one week the commodity will sell for either \(\$ 1\) or \(\$ 4\) an ounce, with these two possibilities being equally likely. (a) If your objective is to maximize the expected amount of money that you possess at the end of the week, what strategy should you employ? (b) If your objective is to maximize the expected amount of the commodity that you possess at the end of the week, what strategy should you employ?

\(A\) and \(B\) play the following game: \(A\) writes down either number 1 or number \(2,\) and \(B\) must guess which one. If the number that \(A\) has written down is \(i\) and \(B\) has guessed correctly, \(B\) receives \(i\) units from \(A .\) If \(B\) makes a wrong guess, \(B\) pays \(\frac{3}{4}\) unit to A. If \(B\) randomizes his decision by guessing 1 with probability \(p\) and 2 with probability \(1-p,\) determine his expected gain if (a) \(A\) has written down number 1 and (b) \(A\) has written down number 2 What value of \(p\) maximizes the minimum possible value of \(B^{\prime}\) s expected gain, and what is this maximin value? (Note that \(B\) 's expected gain depends not only on \(p,\) but also on what \(A\) does. Consider now player \(A .\) Suppose that she also randomizes her decision, writing down number 1 with probability \(q\). What is \(A\) 's expected loss if (c) \(B\) chooses number 1 and \((\text { d) } B \text { chooses number } 2 ?\) What value of \(q\) minimizes \(A\) 's maximum expected loss? Show that the minimum of \(A\) 's maximum expected loss is equal to the maximum of \(B\) 's minimum expected gain. This result, known as the minimax theorem, was first established in generality by the mathematician John von Neumann and is the fundamental result in the mathematical discipline known as the theory of games. The common value is called the value of the game to player \(B\).

Two coins are to be flipped. The first coin will land on heads with probability \(.6,\) the second with probability \(.7 .\) Assume that the results of the flips are independent, and let \(X\) equal the total number of heads that result. (a) Find \(P\\{X=1\\}\) (b) Determine \(E[X]\)

One of the numbers 1 through 10 is randomly chosen. You are to try to guess the number chosen by asking questions with "yes-no" answers. Compute the expected number of questions you will need to ask in each of the following two cases: (a) Your ith question is to be "Is it i?" \(i=\) 1,2,3,4,5,6,7,8,9,10 (b) With each question you try to eliminate onehalf of the remaining numbers, as nearly as possible.

An insurance company writes a policy to the effect that an amount of money \(A\) must be paid if some event \(E\) occurs within a year. If the company estimates that \(E\) will occur within a year with probability \(p,\) what should it charge the customer in order that its expected profit will be 10 percent of \(A ?\)

A sample of 3 items is selected at random from a box containing 20 items of which 4 are defective. Find the expected number of defective items in the sample.

There are two possible causes for a breakdown of a machine. To check the first possibility would cost \(C_{1}\) dollars, and, if that were the cause of the breakdown, the trouble could be repaired at a cost of \(R_{1}\) dollars. Similarly, there are costs \(C_{2}\) and \(R_{2}\) associated with the second possibility. Let \(p\) and \(1-\) \(p\) denote, respectively, the probabilities that the breakdown is caused by the first and second possibilities. Under what conditions on \(p, C_{i}, R_{i}, i=1,2\) should we check the first possible cause of breakdown and then the second, as opposed to reversing the checking order, so as to minimize the expected cost involved in returning the machine to working order? Note: If the first check is negative, we must still check the other possibility.

A person tosses a fair coin until a tail appears for the first time. If the tail appears on the \(n\) th flip, the person wins \(2^{n}\) dollars. Let \(X\) denote the player's winnings. Show that \(E[X]=+\infty .\) This problem is known as the St. Petersburg paradox. (a) Would you be willing to pay \(\$ 1\) million to play this game once? (b) Would you be willing to pay \(\$ 1\) million for each game if you could play for as long as you liked and only had to settle up when you stopped playing?

Each night different meteorologists give us the probability that it will rain the next day. To judge how well these people predict, we will score each of them as follows: If a meteorologist says that it will rain with probability \(p,\) then he or she will receive a score of \(1-(1-p)^{2} \quad\) if it does rain \(1-p^{2} \quad\) if it does not rain We will then keep track of scores over a certain time span and conclude that the meteorologist with the highest average score is the best predictor of weather. Suppose now that a given meteorologist is aware of our scoring mechanism and wants to maximize his or her expected score. If this person truly believes that it will rain tomorrow with probability \(p^{*},\) what value of \(p\) should he or she assert so as to maximize the expected score?

To determine whether they have a certain disease, 100 people are to have their blood tested. However, rather than testing each individual separately, it has been decided first to place the people into groups of \(10 .\) The blood samples of the 10 people in each group will be pooled and analyzed together. If the test is negative, one test will suffice for the 10 people, whereas if the test is positive, each of the 10 people will also be individually tested and, in all, 11 tests will be made on this group. Assume that the probability that a person has the disease is .1 for all people, independently of each other, and compute the expected number of tests necessary for each group. (Note that we are assuming that the pooled test will be positive if at least one person in the pool has the disease.)

A box contains 5 red and 5 blue marbles. Two marbles are withdrawn randomly. If they are the same color, then you win \(\$ 1.10 ;\) if they are different colors, then you win \(-\$ 1.00 .\) (That is, you lose \(\$ 1.00 .\) ) Calculate (a) the expected value of the amount you win; (b) the variance of the amount you win.

If \(E[X]=1\) and \(\operatorname{Var}(X)=5,\) find (a) \(E\left[(2+X)^{2}\right]\) (b) \(\operatorname{Var}(4+3 X)\)

A ball is drawn from an urn containing 3 white and 3 black balls. After the ball is drawn, it is replaced and another ball is drawn. This process goes on indefinitely. What is the probability that, of the first 4 balls drawn, exactly 2 are white?

On a multiple-choice exam with 3 possible answers for each of the 5 questions, what is the probability that a student will get 4 or more correct answers just by guessing?

A man claims to have extrasensory perception. As a test, a fair coin is flipped 10 times and the man is asked to predict the outcome in advance. He gets 7 out of 10 correct. What is the probability that he would have done at least this well if he had no ESP?

Suppose that, in flight, airplane engines will fail with probability \(1-p,\) independently from engine to engine. If an airplane needs a majority of its engines operative to complete a successful flight, for what values of \(p\) is a 5 -engine plane preferable to a 3 -engine plane?

A student is getting ready to take an important oral examination and is concerned about the possibility of having an "on" day or an "off" day. He figures that if he has an on day, then each of his examiners will pass him, independently of each other, with probability \(.8,\) whereas if he has an off day, this probability will be reduced to \(.4 .\) Suppose that the student will pass the examination if a majority of the examiners pass him. If the student feels that he is twice as likely to have an off day as he is to have an on day, should he request an examination with 3 examiners or with 5 examiners?

Suppose that it takes at least 9 votes from a 12 member jury to convict a defendant. Suppose also that the probability that a juror votes a guilty person innocent is \(.2,\) whereas the probability that the juror votes an innocent person guilty is .1. If each juror acts independently and if 65 percent of the defendants are guilty, find the probability that the jury renders a correct decision. What percentage of defendants is convicted?

In some military courts, 9 judges are appointed. However, both the prosecution and the defense attorneys are entitled to a peremptory challenge of any judge, in which case that judge is removed from the case and is not replaced. A defendant is declared guilty if the majority of judges cast votes of guilty, and he or she is declared innocent otherwise. Suppose that when the defendant is, in fact, guilty, each judge will (independently) vote guilty with probability.7, whereas when the defendant is, in fact, innocent, this probability drops to .3. (a) What is the probability that a guilty defendant is declared guilty when there are (i) \(9,(\text { ii) } 8\) and (iii) 7 judges? (b) Repeat part (a) for an innocent defendant. (c) If the prosecution attorney does not exercise the right to a peremptory challenge of a judge, and if the defense is limited to at most two such challenges, how many challenges should the defense attorney make if he or she is 60 percent certain that the client is guilty?

It is known that diskettes produced by a certain company will be defective with probability \(.01,\) independently of each other. The company sells the diskettes in packages of size 10 and offers a money-back guarantee that at most 1 of the 10 diskettes in the package will be defective. The guarantee is that the customer can return the entire package of diskettes if he or she finds more than one defective diskette in it. If someone buys 3 packages, what is the probability that he or she will return exactly 1 of them?

When coin 1 is flipped, it lands on heads with probability \(.4 ;\) when coin 2 is flipped, it lands on heads with probability .7. One of these coins is randomly chosen and flipped 10 times. (a) What is the probability that the coin lands on heads on exactly 7 of the 10 flips? (b) Given that the first of these ten flips lands heads, what is the conditional probability that exactly 7 of the 10 flips land on heads?

Suppose that a biased coin that lands on heads with probability \(p\) is flipped 10 times. Given that a total of 6 heads results, find the conditional probability that the first 3 outcomes are (a) \(h, t, t\) (meaning that the first flip results in heads, the second in tails, and the third in tails); (b) \(t, h, t\)

The expected number of typographical errors on a page of a certain magazine is .2. What is the probability that the next page you read contains (a) 0 and (b) 2 or more typographical errors? Explain your reasoning!

How many people are needed so that the probability that at least one of them has the same birthday as you is greater than \(\frac{1}{2} ?\)

Suppose that the number of accidents occurring on a highway each day is a Poisson random variable with parameter \(\lambda=3\) (a) Find the probability that 3 or more accidents occur today. (b) Repeat part (a) under the assumption that at least 1 accident occurs today.

If you buy a lottery ticket in 50 lotteries, in each of which your chance of winning a prize is \(\frac{1}{100},\) what is the (approximate) probability that you will win a prize (a) at least once? (b) exactly once? (c) at least twice?

The number of times that a person contracts a cold in a given year is a Poisson random variable with parameter \(\lambda=5 .\) Suppose that a new wonder drug (based on large quantities of vitamin \(\mathrm{C}\) ) has just been marketed that reduces the Poisson parameter to \(\lambda=3\) for 75 percent of the population. For the other 25 percent of the population, the drug has no appreciable effect on colds. If an individual tries the drug for a year and has 2 colds in that time, how likely is it that the drug is beneficial for him or her?

Consider \(n\) independent trials, each of which results in one of the outcomes \(1, \ldots, k\) with respective probabilities \(p_{1}, \ldots, p_{k}, \quad \sum_{i=1}^{k} p_{i}=1 .\) Show that if all the \(p_{i}\) are small, then the probability that no trial outcome occurs more than once is approximately equal to \(\exp \left(-n(n-1) \sum_{i} p_{i}^{2} / 2\right)\)

People enter a gambling casino at a rate of 1 every 2 minutes. (a) What is the probability that no one enters between 12: 00 and \(12: 05 ?\) (b) What is the probability that at least 4 people enter the casino during that time?

The suicide rate in a certain state is 1 suicide per 100,000 inhabitants per month. (a) Find the probability that, in a city of 400,000 inhabitants within this state, there will be 8 or more suicides in a given month. (b) What is the probability that there will be at least 2 months during the year that will have 8 or more suicides? (c) Counting the present month as month number \(1,\) what is the probability that the first month to have 8 or more suicides will be month number \(i, i \geq 1 ?\) What assumptions are you making?

Each of 500 soldiers in an army company independently has a certain disease with probability \(1 / 10^{3}\). This disease will show up in a blood test, and to facilitate matters, blood samples from all 500 soldiers are pooled and tested. (a) What is the (approximate) probability that the blood test will be positive (that is, at least one person has the disease)? Suppose now that the blood test yields a positive result. (b) What is the probability, under this circumstance, that more than one person has the disease? One of the 500 people is Jones, who knows that he has the disease. (c) What does Jones think is the probability that more than one person has the disease? Because the pooled test was positive, the authorities have decided to test each individual separately. The first \(i-1\) of these tests were negative, and the \(i\) th one-which was on Jones-was positive. (d) Given the preceding, scenario, what is the probability, as a function of \(i,\) that any of the remaining people have the disease?

A total of \(2 n\) people, consisting of \(n\) married couples, are randomly seated (all possible orderings being equally likely) at a round table. Let \(C_{i}\) denote the event that the members of couple \(i\) are seated next to each other, \(i=1, \ldots, n\) (a) Find \(P\left(C_{i}\right)\) (b) For \(j \neq i,\) find \(P\left(C_{j} | C_{i}\right)\) (c) Approximate the probability, for \(n\) large, that there are no married couples who are seated next to each other.

At time \(0,\) a coin that comes up heads with probability \(p\) is flipped and falls to the ground. Suppose it lands on heads. At times chosen according to a Poisson process with rate \(\lambda,\) the coin is picked up and flipped. (Between these times the coin remains on the ground.) What is the probability that the coin is on its head side at time \(t ?\) Hint What would be the conditional probability if there were no additional flips by time \(t,\) and what would it be if there were additional flips by time \(t ?\)

Consider a roulette wheel consisting of 38 numbers 1 through \(36,0,\) and double \(0 .\) If Smith always bets that the outcome will be one of the numbers 1 through \(12,\) what is the probability that (a) Smith will lose his first 5 bets; (b) his first win will occur on his fourth bet?

Two athletic teams play a series of games; the first team to win 4 games is declared the overall winner. Suppose that one of the teams is stronger than the other and wins each game with probability .6 independently of the outcomes of the other games. Find the probability, for \(i=4,5,6,7,\) that the stronger team wins the series in exactly \(i\) games. Compare the probability that the stronger team wins with the probability that it would win a 2 -outof-3 series.