2.5 Supplementary Exercises

Exercise 2.5.2¶

Suppose that a fair coin is tossed repeatedly and independently until both a head and a tail have appeared at least once.

(a) Describe the sample space of this experiment. (b) What is the probability that exactly three tosses will be required?

Exercise 2.5.3¶

Suppose that $A$ and $B$ are events such that $\Pr(A) = 1/3$, $\Pr(B) = 1/5$, and $\Pr(A \mid B) + \Pr(B \mid A) = 2/3$. Evaluate $\Pr(A^c \cup B^c)$.

Exercise 2.5.4¶

Suppose that $A$ and $B$ are independent events such that $\Pr(A) = 1/3$ and $\Pr(B) > 0$. What is the value of $\Pr(A \cup B^c \mid B)$?

Exercise 2.5.5¶

Suppose that in 10 rolls of a balanced die, the number 6 appeared exactly three times. What is the probability that the first three rolls each yielded the number 6?

Exercise 2.5.6¶

Suppose that $A$, $B$, and $D$ are events such that $A$ and $B$ are independent, $\Pr(A \cap B \cap D) = 0.04$, $\Pr(D \mid A \cap B) = 0.25$, and $\Pr(B) = 4 \Pr(A)$. Evaluate $\Pr(A \cup B)$.

Exercise 2.5.7¶

Suppose that the events $A$, $B$, and $C$ are mutually independent. Under what conditions are $A^c$, $B^c$, and $C^c$ mutually independent?

Exercise 2.5.8¶

Suppose that the events $A$ and $B$ are disjoint and that each has positive probability. Are $A$ and $B$ independent?

Exercise 2.5.9¶

Suppose that $A$, $B$, and $C$ are three events such that $A$ and $B$ are disjoint, $A$ and $C$ are independent, and $B$ and $C$ are independent. Suppose also that $4\Pr(A) = 2\Pr(B) = \Pr(C) > 0$ and $\Pr(A \cup B \cup C) = 5\Pr(A)$. Determine the value of $\Pr(A)$.

Exercise 2.5.10¶

Suppose that each of two dice is loaded so that when either die is rolled, the probability that the number $k$ will appear is 0.1 for $k = 1$, 2, 5, or 6 and is 0.3 for $k = 3$ or 4. If the two loaded dice are rolled independently, what is the probability that the sum of the two numbers that appear will be 7?

Exercise 2.5.11¶

Suppose that there is a probability of $1/50$ that you will win a certain game. If you play the game 50 times, independently, what is the probability that you will win at least once?

Exercise 2.5.12¶

Suppose that a balanced die is rolled three times, and let $X_i$ denote the number that appears on the ith roll ($i = 1, 2, 3$). Evaluate $\Pr(X_1 > X_2 > X_3)$.

Exercise 2.5.13¶

Three students $A$, $B$, and $C$ are enrolled in the same class. Suppose that $A$ attends class 30 percent of the time, $B$ attends class 50 percent of the time, and $C$ attends class 80 percent of the time. If these students attend class independently of each other, what is (a) the probability that at least one of them will be in class on a particular day and (b) the probability that exactly one of them will be in class on a particular day?

Exercise 2.5.14¶

Consider the World Series of baseball, as described in Div of 2.2 Independent Events. If there is probability $p$ that team $A$ will win any particular game, what is the probability that it will be necessary to play seven games in order to determine the winner of the Series?

Exercise 2.5.15¶

Suppose that three red balls and three white balls are thrown at random into three boxes and and that all throws are independent. What is the probability that each box contains one red ball and one white ball?

Exercise 2.5.16¶

If five balls are thrown at random into $n$ boxes, and all throws are independent, what is the probability that no box contains more than two balls?

Exercise 2.5.17¶

Bus tickets in a certain city contain four numbers, $U$, $V$, $W$, and $X$. Each of these numbers is equally likely to be any of the 10 digits $0, 1, \ldots, 9$, and the four numbers are chosen independently. A bus rider is said to be lucky if $U + V = W + X$. What proportion of the riders are lucky?

Exercise 2.5.18¶

A certain group has eight members. In January, three members are selected at random to serve on a committee. In February, four members are selected at random and independently of the first selection to serve on another committee. In March, five members are selected at random and independently of the previous two selections to serve on a third committee. Determine the probability that each of the eight members serves on at least one of the three committees.

Exercise 2.5.19¶

For the conditions of Div, determine the probability that two particular members $A$ and $B$ will serve together on at least one of the three committees.

Exercise 2.5.20¶

Suppose that two players $A$ and $B$ take turns rolling a pair of balanced dice and that the winner is the first player who obtains the sum of 7 on a given roll of the two dice. If $A$ rolls first, what is the probability that $B$ will win?

Exercise 2.5.21¶

Three players $A$, $B$, and $C$ take turns tossing a fair coin. Suppose that $A$ tosses the coin first, $B$ tosses second, and $C$ tosses third; and suppose that this cycle is repeated indefinitely until someone wins by being the first player to obtain a head. Determine the probability that each of three players will win.

Exercise 2.5.22¶

Suppose that a balanced die is rolled repeatedly until the same number appears on two successive rolls, and let $X$ denote the number of rolls that are required. Determine the value of $\Pr(X = x)$, for $x = 2, 3, \ldots$.

Exercise 2.5.23¶

Suppose that 80 percent of all statisticians are shy, whereas only 15 percent of all economists are shy. Suppose also that 90 percent of the people at a large gathering are economists and the other 10 percent are statisticians. If you meet a shy person at random at the gathering, what is the probability that the person is a statistician?

Exercise 2.5.24¶

Dreamboat cars are produced at three different factories $A$, $B$, and $C$. Factory $A$ produces 20 percent of the total output of Dreamboats, $B$ produces 50 percent, and $C$ produces 30 percent. However, 5 percent of the cars produced at $A$ are lemons, 2 percent of those produced at $B$ are lemons, and 10 percent of those produced at $C$ are lemons. If you buy a Dreamboat and it turns out to be a lemon, what is the probability that it was produced at factory $A$?

Exercise 2.5.25¶

Suppose that 30 percent of the bottles produced in a certain plant are defective. If a bottle is defective, the probability is 0.9 that an inspector will notice it and remove it from the filling line. If a bottle is not defective, the probability is 0.2 that the inspector will think that it is defective and remove it from the filling line.

a. If a bottle is removed from the filling line, what is the probability that it is defective? b. If a customer buys a bottle that has not been removed from the filling line, what is the probability that it is defective?

Exercise 2.5.26¶

Suppose that a fair coin is tossed until a head is obtained and that this entire experiment is then performed independently a second time. What is the probability that the second experiment requires more tosses than the first experiment?

Exercise 2.5.27¶

Suppose that a family has exactly $n$ children ($n \geq 2$). Assume that the probability that any child will be a girl is $1/2$ and that all births are independent. Given that the family has at least one girl, determine the probability that the family has at least one boy.

Exercise 2.5.28¶

Suppose that a fair coin is tossed independently $n$ times. Determine the probability of obtaining exactly $n - 1$ heads, given (a) that at least $n − 2$ heads are obtained and (b) that heads are obtained on the first $n − 2$ tosses.

Exercise 2.5.29¶

Suppose that 13 cards are selected at random from a regular deck of 52 playing cards.

a. If it is known that at least one ace has been selected, what is the probability that at least two aces have been selected? b. If it is known that the ace of hearts has been selected, what is the probability that at least two aces have been selected?

Exercise 2.5.30¶

Suppose that $n$ letters are placed at random in $n$ envelopes, as in the matching problem of 1.10 The Probability of a Union of Events, and let $q_n$ denote the probability that no letter is placed in the correct envelope. Show that the probability that exactly one letter is placed in the correct envelope is $q_{n−1}$.

Exercise 2.5.31¶

Consider again the conditions of Div. Show that the probability that exactly two letters are placed in the correct envelopes is $(1/2)q_{n−2}$.

Exercise 2.5.32¶

Consider again the conditions of Div of 2.2 Independent Events. If exactly one of the two students $A$ and $B$ is in class on a given day, what is the probability that it is $A$?

Exercise 2.5.33¶

Consider again the conditions of exr-1-10-2 of 1.10 The Probability of a Union of Events. If a family selected at random from the city subscribes to exactly one of the three newspapers $A$, $B$, and $C$, what is the probability that it is $A$?

Exercise 2.5.34¶

Three prisoners $A$, $B$, and $C$ on death row know that exactly two of them are going to be executed, but they do not know which two. Prisoner $A$ knows that the jailer will not tell him whether or not he is going to be executed. He therefore asks the jailer to tell him the name of one prisoner other than $A$ himself who will be executed. The jailer responds that $B$ will be executed. Upon receiving this response, Prisoner $A$ reasons as follows: Before he spoke to the jailer, the probability was $2/3$ that he would be one of the two prisoners executed. After speaking to the jailer, he knows that either he or prisoner $C$ will be the other one to be executed. Hence, the probability that he will be executed is now only $1/2$. Thus, merely by asking the jailer his question, the prisoner reduced the probability that he would be executed from $2/3$ to $1/2$, because he could go through exactly this same reasoning regardless of which answer the jailer gave. Discuss what is wrong with prisoner $A$’s reasoning.

Exercise 2.5.35¶

Suppose that each of two gamblers $A$ and $B$ has an initial fortune of 50 dollars, and that there is probability $p$ that gambler $A$ will win on any single play of a game against gambler $B$. Also, suppose either that one gambler can win one dollar from the other on each play of the game or that they can double the stakes and one can win two dollars from the other on each play of the game. Under which of these two conditions does $A$ have the greater probability of winning the initial fortune of $B$ before losing her own for each of the following conditions: (a) $p < 1/2$; (b) $p > 1/2$; (c) $p = 1/2$?

Exercise 2.5.36¶

A sequence of $n$ job candidates is prepared to interview for a job. We would like to hire the best candidate, but we have no information to distinguish the candidates before we interview them. We assume that the best candidate is equally likely to be each of the $n$ candidates in the sequence before the interviews start. After the interviews start, we are able to rank those candidates we have seen, but we have no information about where the remaining candidates rank relative to those we have seen. After each interview, it is required that either we hire the current candidate immediately and stop the interviews, or we must let the current candidate go and we never can call them back. We choose to interview as follows: We select a number $0 \leq r < n$ and we interview the first $r$ candidates without any intention of hiring them. Starting with the next candidate $r + 1$, we continue interviewing until the current candidate is the best we have seen so far. We then stop and hire the current candidate. If none of the candidates from $r + 1$ to $n$ is the best, we just hire candidate $n$. We would like to compute the probability that we hire the best candidate and we would like to choose $r$ to make this probability as large as possible. Let $A$ be the event that we hire the best candidate, and let $B_i$ be the event that the best candidate is in position $i$ in the sequence of interviews.

a. Let $i > r$. Find the probability that the candidate who is relatively the best among the first $i$ interviewed appears in the first $r$ interviews. b. Prove that $\Pr(A \mid B_i) = 0$ for $i \leq r$ and $\Pr(A \mid B_i) = r/(i − 1)$ for $i > r$. c. For fixed $r$, let $p_r$ be the probability of $A$ using that value of $r$. Prove that $p_r = (r/n)\sum_{i=r+1}^{n}(i-1)^{-1}$. d. Let $q_r = p_r − p_{r−1}$ for $r = 1, \ldots, n - 1$, and prove that $q_r$ is a strictly decreasing function of $r$. e. Show that a value of $r$ that maximizes $p_r$ is the last $r$ such that $q_r > 0$. (Hint: Write $p_r = p_0 + q_1 + \cdots + q_r$ for $r > 0$.) f. For $n = 10$, find the value of $r$ that maximizes $p_r$, and find the corresponding $p_r$ value.