Chapter 7

A player throws a fair die and simultaneously flips a fair coin. If the coin lands heads, then she wins twice, and if tails, then one-half of the value that appears on the die. Determine her expected winnings.

The game of Clue involves 6 suspects, 6 weapons, and 9 rooms. One of each is randomly chosen and the object of the game is to guess the chosen three.(a) How many solutions are possible? In one version of the game, the selection is made and then each of the players is randomly given three of the remaining cards. Let \(S, W\) and \(R\) be, respectively, the numbers of suspects, weapons, and rooms in the set of three cards given to a specified player. Also, let \(X\) denote the number of solutions that are possible after that player observes his or her three cards. (b) Express \(X\) in terms of \(S, W,\) and \(R\) (c) Find \(E[X]\)

Gambles are independent, and each one results in the player being equally likely to win or lose 1 unit. Let \(W\) denote the net winnings of a gambler whose strategy is to stop gambling immediately after his first win. Find (a) \(P(W>0\\}\) (b) \(P\\{W<0\\}\) (c) \(E[W]\)

If \(X\) and \(Y\) have joint density function $$f_{X, Y}(x, y)=\left\\{\begin{array}{ll} 1 / y, & \text { if } 0

A fair die is rolled 10 times. Calculate the expected sum of the 10 rolls.

Suppose that \(A\) and \(B\) each randomly and independently choose 3 of 10 objects. Find the expected number of objects (a) chosen by both \(A\) and \(B\) (b) not chosen by either \(A\) or \(B\) (c) chosen by exactly one of \(A\) and \(B\)

\(N\) people arrive separately to a professional dinner. Upon arrival, each person looks to see if he or she has any friends among those present. That person then sits either at the table of a friend or at an unoccupied table if none of those present is a friend. Assuming that each of the \(\left(\begin{array}{l}N \\ 2\end{array}\right)\)pairs of people is, independently, a pair of friends with probability \(p,\) find the expected number of occupied tables. Hint: Let \(X_{i}\) equal 1 or \(0,\) depending on whether the \(i\) th arrival sits at a previously unoccupied table.

Consider 3 trials, each having the same probability of success. Let \(X\) denote the total number of successes in these trials. If \(E[X]=1.8\) what is (a) the largest possible value of \(P\\{X=3\\} ?\) (b) the smallest possible value of \(P\\{X=3\\} ?\) In both cases, construct a probability scenario that results in \(P\\{X=3\\}\) having the stated value. Hint: For part (b), you might start by letting \(U\) be a uniform random variable on (0,1) and then defining the trials in terms of the value of \(U\)

Consider \(n\) independent flips of a coin having probability \(p\) of landing on heads. Say that a changeover occurs whenever an outcome differs from the one preceding it. For instance, if \(n=\) 5 and the outcome is \(H H T H T,\) then there are 3 changeovers. Find the expected number of changeovers. Hint: Express the number of changeovers as the sum of \(n-1\) Bernoulli random variables.

A group of \(n\) men and \(n\) women is lined up at random. (a) Find the expected number of men who have a woman next to them. (b) Repeat part (a), but now assuming that the group is randomly seated at a round table.

A set of 1000 cards numbered 1 through 1000 is randomly distributed among 1000 people with each receiving one card. Compute the expected number of cards that are given to people whose age matches the number on the card.

Let \(Z\) be a standard normal random variable, and, for a fixed \(x,\) set $$ X=\left\\{\begin{array}{ll} Z & \text { if } Z>x \\ 0 & \text { otherwise } \end{array}\right. $$ Show that \(E[X]=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2}\)

Cards from an ordinary deck of 52 playing cards are turned face up one at a time. If the 1 st card is an ace, or the 2 nd a deuce, or the 3 rd a three, or \(\dots,\) or the 13 th a king, or the 14 an ace, and so on, we say that a match occurs. Note that we do not require that the \((13 n+1)\) th card be any particular ace for a match to occur but only that it be an ace. Compute the expected number of matches that occur.

A certain region is inhabited by \(r\) distinct types of a certain species of insect. Each insect caught will, independently of the types of the previous catches, be of type \(i\) with probability $$ P_{i}, i=1, \ldots, r \quad \sum_{1}^{r} P_{i}=1 $$ (a) Compute the mean number of insects that are caught before the first type 1 catch. (b) Compute the mean number of types of insects that are caught before the first type 1 catch.

In an urn containing \(n\) balls, the \(i\) th ball has weight \(W(i), i=1, \ldots, n .\) The balls are removed without replacement, one at a time, according to the following rule: At each selection, the probability that a given ball in the urn is chosen is equal to its weight divided by the sum of the weights remaining in the urn. For instance, if at some time \(i_{1}, \ldots, i_{r}\) is the set of balls remaining in the urn, then the next selection will be \(i_{j}\) with probability \(W\left(i_{j}\right) / \sum_{k=1}^{r} W\left(i_{k}\right), j=1, \ldots, r .\) Compute the expected number of balls that are withdrawn before ball number 1 is removed.

How many times would you expect to roll a fair die before all 6 sides appeared at least once?

Urn 1 contains 5 white and 6 black balls, while urn 2 contains 8 white and 10 black balls. Two balls are randomly selected from urn 1 and are put into urn \(2 .\) If 3 balls are then randomly selected from urn \(2,\) compute the expected number of white balls in the trio. Hint: Let \(X_{i}=1\) if the \(i\) th white ball initially in urn 1 is one of the three selected, and let \(X_{i}=0\) otherwise. Similarly, let \(Y_{i}=1\) if the \(i\) th white ball from urn 2 is one of the three sclected, and let \(Y_{i}=0\) otherwise. The number of white balls in the trio can now be written as \(\sum_{1}^{J} X_{i}+\sum_{1}^{\delta} Y_{i}\)

A bottle initially contains \(m\) large pills and \(n\) small pills. Each day, a patient randomly chooses one of the pills. If a small pill is chosen, then that pill is eaten. If a large pill is chosen, then the pill is broken in two; one part is returned to the bottle (and is now considered a small pill) and the other part is then eaten. (a) Let \(X\) denote the number of small pills in the bottle after the last large pill has been chosen and its smaller half returned. Find \(E[X]\) Hint: Define \(n+m\) indicator variables, one for each of the small pills initially present and one for each of the \(m\) small pills created when a large one is split in two. Now use the argument of Example \(2 \mathrm{m}\) (b) Let \(Y\) denote the day on which the last large pill is chosen. Find \(E[Y]\) Hint: What is the relationship between \(X\) and \(Y ?\)

Let \(X_{1}, X_{2}, \ldots\) be a sequence of independent and identically distributed continuous random variables. Let \(N \geq 2\) be such that $$ X_{1} \geq X_{2} \geq \dots \geq X_{N-1}

If \(X_{1}, X_{2}, \ldots, X_{n}\) are independent and identically distributed random variables having uniform distributions over \((0,1),\) find (a) \(E\left[\max \left(X_{1}, \ldots, X_{n}\right)\right]\) (b) \(E\left[\min \left(X_{1}, \ldots, X_{n}\right)\right]\)

If 101 items are distributed among 10 boxes, then at least one of the boxes must contain more than 10 items. Use the probabilistic method to prove this result.

The \(k\) -of-r-out-of- \(n\) circular reliability system, \(k \leq\) \(r \leqq n,\) consists of \(n\) components that are arranged in a circular fashion. Each component is either functional or failed, and the system functions if there is no block of \(r\) consecutive components of which at least \(k\) are failed. Show that there is no way to arrange 47 components, 8 of which are failed, to make a functional 3 -of- 12 -out-of- 47 circular system.

There are 4 different types of coupons, the first 2 of which compose one group and the second 2 another group. Each new coupon obtained is type \(i\) with probability \(p_{i},\) where \(p_{1}=p_{2}=1 / 8, p_{3}=\) \(p_{4}=3 / 8 .\) Find the expected number of coupons that one must obtain to have at least one of (a) all 4 types; (b) all the types of the first group; (c) all the types of the second group; (d) all the types of either group.

If \(X\) and \(Y\) are independent and identically distributed with mean \(\mu\) and variance \(\sigma^{2},\) find $$ E\left[(X-Y)^{2}\right] $$

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

Cards from an ordinary deck are turned face up one at a time. Compute the expected number of cards that need to be turned face up in order to obtain (a) 2 aces; (b) 5 spades; (c) all 13 hearts.

A die is rolled twice. Let \(X\) equal the sum of the outcomes, and let \(Y\) equal the first outcome minus the second. Compute \(\operatorname{Cov}(X, Y)\)

Let \(X_{1}, \ldots\) be independent with common mean \(\mu\) and common variance \(\sigma^{2},\) and set \(Y_{n}=X_{n}+\) \(X_{n+1}+X_{n+2} .\) For \(j \geq 0,\) find \(\operatorname{Cov}\left(Y_{n}, Y_{n+j}\right)\)

The joint density function of \(X\) and \(Y\) is given by $$ f(x, y)=\frac{1}{y} e^{-(y+x / y)}, \quad x>0, y>0 $$ Find \(E[X], E[Y],\) and show that \(\operatorname{Cov}(X, Y)=1\)

A pond contains 100 fish, of which 30 are carp. If 20 fish are caught, what are the mean and variance of the number of carp among the \(20 ?\) What assumptions are you making?

A group of 20 people consisting of 10 men and 10 women is randomly arranged into 10 pairs of 2 each. Compute the expectation and variance of the number of pairs that consist of a man and a woman. Now suppose the 20 people consist of 10 married couples. Compute the mean and variance of the number of married couples that are paired together.

Between two distinct methods for manufacturing certain goods, the quality of goods produced by method \(i\) is a continuous random variable having distribution \(F_{i}, i=1,2 .\) Suppose that \(n\) goods are produced by method 1 and \(m\) by method \(2 .\) Rank the \(n+m\) goods according to quality, and let $$ X_{j}=\left\\{\begin{array}{ll} 1 & \text { if the } j \text { th best was produced from } \\ & \text { method } 1 \\ 2 & \text { otherwise } \end{array}\right. $$ For the vector \(X_{1}, X_{2}, \ldots, X_{n+m},\) which consists of \(n\) 1's and \(m\) 2's, let \(R\) denote the number of runs of \(1 .\) For instance, if \(n=5, m=2,\) and \(X=\) \(1,2,1,1,1,1,2,\) then \(R=2 .\) If \(F_{1}=F_{2}\) (that is, if the two methods produce identically distributed goods), what are the mean and variance of \(R ?\)

If \(X_{1}, X_{2}, X_{3},\) and \(X_{4}\) are (pairwise) uncorrelated random variables, each having mean 0 and variance \(1,\) compute the correlations of (a) \(X_{1}+X_{2}\) and \(X_{2}+X_{3}\) (b) \(X_{1}+X_{2}\) and \(X_{3}+X_{4}\)

Consider the following dice game, as played at a certain gambling casino: Players 1 and 2 roll a pair of dice in turn. The bank then rolls the dice to determine the outcome according to the following rule: Player \(i, i=1,2,\) wins if his roll is strictly greater than the bank's. For \(i=1,2,\) let $$ I_{i}=\left\\{\begin{array}{ll} 1 & \text { if } i \text { wins } \\ 0 & \text { otherwise } \end{array}\right. $$ and show that \(I_{1}\) and \(I_{2}\) are positively correlated. Explain why this result was to be expected.

Consider a graph having \(n\) vertices labeled \(1,2, \ldots, n,\) and suppose that, between each of the \(\left(\begin{array}{l}n \\ 2\end{array}\right)\) pairs of distinct vertices, an edge is independently present with probability \(p .\) The degree of vertex \(i,\) designated as \(D_{i}\), is the number of edges that have vertex \(i\) as one of their vertices. (a) What is the distribution of \(D_{i} ?\) (b) Find \(\rho\left(D_{i}, D_{j}\right),\) the correlation between \(D_{i}\) and \(D_{j}\)

A fair die is successively rolled. Let \(X\) and \(Y\) denote, respectively, the number of rolls necessary to obtain a 6 and a \(5 .\) Find (a) \(E[X]\) (b) \(E[X | Y=1]\) (c) \(E[X | Y=5]\)

There are two misshapen coins in a box; their probabilities for landing on heads when they are flipped are, respectively, 4 and \(.7 .\) One of the coins is to be randomly chosen and flipped 10 times. Given that two of the first three flips landed on heads, what is the conditional expected number of heads in the 10 flips?

The joint density of \(X\) and \(Y\) is given by \(f(x, y)=\frac{e^{-x / y} e^{-y}}{y}, \quad 0

A population is made up of \(r\) disjoint subgroups. Let \(p_{i}\) denote the proportion of the population that is in subgroup \(i, i=1, \ldots, r .\) If the average weight of the members of subgroup \(i\) is \(w_{i}, i=1, \ldots, r\) what is the average weight of the members of the population?

A prisoner is trapped in a cell containing 3 doors. The first door leads to a tunnel that returns him to his cell after 2 days' travel. The second leads to a tunnel that returns him to his cell after 4 days' travel. The third door leads to freedom after 1 day of travel. If it is assumed that the prisoner will always select doors \(1,2,\) and 3 with respective probabilities \(.5,3,\) and \(.2,\) what is the expected number of days until the prisoner reaches freedom?

Ten hunters are waiting for ducks to fly by. When a flock of ducks flies overhead, the hunters fire at the same time, but each chooses his target at random, independently of the others. If each hunter independently hits his target with probability .6 compute the expected number of ducks that are hit. Assume that the number of ducks in a flock is a Poisson random variable with mean 6

Suppose that the expected number of accidents per week at an industrial plant is \(5 .\) Suppose also that the numbers of workers injured in each accident are independent random variables with a common mean of \(2.5 .\) If the number of workers injured in each accident is independent of the number of accidents that occur, compute the expected number of workers injured in a week.

A coin having probability \(p\) of coming up heads is continually flipped until both heads and tails have appeared. Find (a) the expected number of flips; (b) the probability that the last flip lands on heads.

There are \(n+1\) participants in a game. Each person independently is a winner with probability \(p .\) The winners share a total prize of 1 unit. (For instance, if 4 people win, then cach of them receives \(\frac{1}{4},\) whereas if there are no winners, then none of the participants receive anything.) Let \(A\) denote a specificd one of the players, and let \(X\) denote the amount that is received by \(A\)(a) Compute the expected total prize shared by the players. (b) Argue that \(E[X]=\frac{1-(1-p)^{n+1}}{n+1}\) (c) Compute \(E[X]\) by conditioning on whether \(A\) is a winner, and conclude that $$ E\left[(1+B)^{-1}\right]=\frac{1-(1-p)^{n+1}}{(n+1) p} $$ when \(B\) is a binomial random variable with parameters \(n\) and \(p\)

Each of \(m+2\) players pays 1 unit to a kitty in order to play the following game: A fair coin is to be flipped successively \(n\) times, where \(n\) is an odd number, and the successive outcomes are noted. Before the \(n\) llips, each player writes down a prediction of the outcomes. For instance, if \(n=3\) then a player might write down \((H, H, T),\) which means that he or she predicts that the first flip will land on heads, the second on heads, and the third on tails. After the coins are flipped, the players count their total number of correct predictions. Thus, if the actual outcomes are all heads, then the player who wrote \((H, H, T)\) would have 2 correct predictions. The total kitty of \(m+2\) is then evenly split up among those players having the largest number of correct predictions. since each of the coin flips is equally likely to land on either heads or tails, \(m\) of the players have decided to make their predictions in a totally random fashion. Specifically, they will each flip one of their own fair coins \(n\) times and then use the result as their prediction. However, the final 2 of the players have formed a syndicate and will use the following strategy: One of them will make predictions in the same random fashion as the other \(m\) players, but the other one will then predict exactly the opposite of the first. That is, when the randomizing member of the syndicate predicts an \(H,\) the other member predicts a \(T .\) For instance, if the randomizing member of the syndicate predicts \((H, H, T),\) then the other one predicts \((T,\) \(T, H)\) (a) Argue that exactly one of the syndicate members will have more than \(n / 2\) correct predictions. (Remember, \(n\) is odd.) (b) Let \(X\) denote the number of the \(m\) nonsyndicate players that have more than \(n / 2\) correct predictions. What is the distribution of \(X ?\) (c) With \(X\) as defined in part (b), argue that \(E[\text { payoff to the syndicate }]=(m+2)\) $$ \times E\left[\frac{1}{X+1}\right] $$(d) Use part (c) of Problem 59 to conclude that \(\begin{aligned} E[\text { payoff to the syndicate }]=& \frac{2(m+2)}{m+1} \\\ & \times\left[1-\left(\frac{1}{2}\right)^{m+1}\right] \end{aligned}\) and explicitly compute this number when \(m=\) \(1,2,\) and \(3 .\) Because it can be shown that $$ \frac{2(m+2)}{m+1}\left[1-\left(\frac{1}{2}\right)^{m+1}\right]>2 $$ it follows that the syndicate's strategy always gives it a positive expected profit.

Let \(X_{1}, \ldots\) be independent random variables with the common distribution function \(F,\) and suppose they are independent of \(N,\) a geometric random variable with parameter \(p .\) Let \(M=\) \(\max \left(X_{1}, \ldots, X_{N}\right)\) (a) Find \(P\\{M \leq x\\}\) by conditioning on \(N\) (b) Find \(P\\{M \leq x | N=1\\}\) (c) Find \(P(M \leq x | N>1\\}\) (d) Use (b) and (c) to rederive the probability you found in (a).

Let \(U_{1}, U_{2}, \ldots\) be a sequence of independent uniform (0,1) random variables. In Example \(5 \mathrm{i}\) we showed that, for \(0 \leq x \leq 1, E[N(x)]=e^{x},\) where $$ N(x)=\min \left\\{n: \sum_{i=1}^{n} U_{i}>x\right\\} $$ This problem gives another approach to establishing that result. (a) Show by induction on \(n\) that, for \(0

An urn contains 30 balls, of which 10 are red and 8 are blue. From this urn, 12 balls are randomly withdrawn. Let \(X\) denote the number of red and \(Y\) the number of blue balls that are withdrawn. Find \(\operatorname{Cov}(X, Y)\) (a) by defining appropriate indicator (that is, Bernoulli) random variables $$ X_{i}, Y_{j} \text { such that } X=\sum_{i=1}^{10} X_{i}, Y=\sum_{j=1}^{8} Y_{j} $$ (b) by conditioning (on either \(X\) or \(Y\) ) to determine \(E[X Y]\)

Type \(i\) light bulbs function for a random amount of time having mean \(\mu_{i}\) and standard deviation \(\sigma_{i}, i=1,2 .\) A light bulb randomly chosen from a bin of bulbs is a type 1 bulb with probability \(p\) and a type 2 bulb with probability \(1-p .\) Let \(X\) denote the lifetime of this bulb. Find (a) \(E[X]\) (b) \(\operatorname{Var}(X)\)

In Example \(5 \mathrm{c},\) compute the variance of the length of time until the miner reaches safety.