Chapter 3

In a class, there are 4 freshman boys, 6 freshman girls, and 6 sophomore boys. How many sophomore girls must be present if sex and class are to be independent when a student is selected at random?

Suppose that you continually collect coupons and that there are \(m\) different types. Suppose also that each time a new coupon is obtained, it is a type i coupon with probability \(p_{i}, i=1, \ldots, m .\) Suppose that you have just collected your \(n\)th coupon. What is the probability that it is a new type? Hint: Condition on the type of this coupon.

A simplified model for the movement of the price of a stock supposes that on each day the stock's price either moves up 1 unit with probability \(p\) or moves down 1 unit with probability \(1-p .\) The changes on different days are assumed to be independent. (a) What is the probability that after 2 days the stock will be at its original price? (b) What is the probability that after 3 days the stock's price will have increased by 1 unit? (c) Given that after 3 days the stock's price has increased by 1 unit, what is the probability that it went up on the first day?

Suppose that we want to generate the outcome of the flip of a fair coin, but that all we have at our disposal is a biased coin which lands on heads with some unknown probability \(p\) that need not be equal to \(\frac{1}{2} .\) Consider the following procedure for accomplishing our task: 1\. Flip the coin. 2\. Flip the coin again. 3\. If both flips land on heads or both land on tails, return to step 1. 4\. Let the result of the last flip be the result of the experiment. (a) Show that the result is equally likely to be either heads or tails. (b) Could we use a simpler procedure that continues to flip the coin until the last two flips are different and then lets the result be the outcome of the final flip?

Independent flips of a coin that lands on heads with probability \(p\) are made. What is the probability that the first four outcomes are (a) \(H, H, H, H ?\) (b) \(T, H, H, H ?\) (c) What is the probability that the pattern \(T, H\) \(H, H\) occurs before the pattern \(H, H, H, H ?\) Hint for part \((c):\) How can the pattern \(H, H, H, H\) occur first?

The color of a person's eyes is determined by a single pair of genes. If they are both blue-eyed genes, then the person will have blue eyes; if they are both brown-eyed genes, then the person will have brown eyes; and if one of them is a blue-eyed gene and the other a brown-eyed gene, then the person will have brown eyes. (Because of the latter fact, we say that the brown-eyed gene is dominant over the blue-eyed one.) A newborn child independently receives one eye gene from each of its parents, and the gene it receives from a parent is equally likely to be either of the two eye genes of that parent. Suppose that Smith and both of his parents have brown eyes, but Smith's sister has blue eyes. (a) What is the probability that Smith possesses a blue-eyed gene? (b) Suppose that Smith's wife has blue eyes. What is the probability that their first child will have blue eyes? (c) If their first child has brown eyes, what is the probability that their next child will also have brown eyes?

Genes relating to albinism are denoted by \(A\) and a. Only those people who receive the \(a\) gene from both parents will be albino. Persons having the gene pair \(A, a\) are normal in appearance and, because they can pass on the trait to their offspring, are called carriers. Suppose that a normal couple has two children, exactly one of whom is an albino. Suppose that the nonalbino child mates with a person who is known to be a carrier for albinism. (a) What is the probability that their first offspring is an albino? (b) What is the conditional probability that their second offspring is an albino given that their firstborn is not?

Barbara and Dianne go target shooting. Suppose that each of Barbara's shots hits a wooden duck target with probability \(p_{1},\) while each shot of Dianne's hits it with probability \(p_{2} .\) Suppose that they shoot simultaneously at the same target. If the wooden duck is knocked over (indicating that it was hit), what is the probability that (a) both shots hit the duck? (b) Barbara's shot hit the duck? What independence assumptions have you made?

\(A\) and \(B\) are involved in a duel. The rules of the duel are that they are to pick up their guns and shoot at each other simultaneously. If one or both are hit, then the duel is over. If both shots miss, then they repeat the process. Suppose that the results of the shots are independent and that each shot of \(A\) will hit \(B\) with probability \(p_{A},\) and each shot of \(B\) will hit \(A\) with probability \(p_{B}\). What is (a) the probability that \(A\) is not hit? (b) the probability that both duelists are hit? (c) the probability that the duel ends after the \(n\) th round of shots? (d) the conditional probability that the duel ends after the \(n\) th round of shots given that \(A\) is not hit? (e) the conditional probability that the duel ends after the \(n\) th round of shots given that both duelists are hit?

An engineering system consisting of \(n\) components is said to be a \(k\) -out- of- \(n\) system \((k \leq n)\) if the system functions if and only if at least \(k\) of the \(n\) components function. Suppose that all components function independently of one another. (a) If the \(i\) th component functions with probability \(P_{i}, i=\) \(1,2,3,4,\) compute the probability that a 2 -out-of- 4 system functions. (b) Repeat part (a) for a 3 -out-of- 5 system. (c) Repeat for a \(k\) -out-of- \(n\) system when all the \(P_{i}\) equal \(p\) (that is, \(\left.P_{i}=p, i=1,2, \ldots, n\right)\)

In Problem \(3.65 \mathrm{a},\) find the conditional probability that relays 1 and 2 are both closed given that a current flows from \(A\) to \(B.\)

A certain organism possesses a pair of each of 5 different genes (which we will designate by the first 5 letters of the English alphabet). Each gene appears in 2 forms (which we designate by lowercase and capital letters). The capital letter will be assumed to be the dominant gene, in the sense that if an organism possesses the gene pair \(x X\) then it will outwardly have the appearance of the \(X\) gene. For instance, if \(X\) stands for brown eyes and \(x\) for blue eyes, then an individual having either gene pair \(X X\) or \(x X\) will have brown eyes, whereas one having gene pair \(x x\) will have blue eyes. The characteristic appearance of an organism is called its phenotype, whereas its genetic constitution is called its genotype. (Thus, 2 organisms with respective genotypes \(a A, b B, c c, d D\) ee and \(A A, B B, c c, D D,\) ee would have different genotypes but the same phenotype.) In a mating between 2 organisms, each one contributes, at random, one of its gene pairs of each type. The 5 contributions of an organism (one of each of the 5 types) are assumed to be independent and are also independent of the contributions of the organism's mate. In a mating between organisms having genotypes \(a A, b B, c C, d D, e E\) and \(a a, b B, c c\) \(D d,\) ee what is the probability that the progeny will (i) phenotypically and (ii) genotypically resemble (a) the first parent? (b) the second parent? (c) either parent? (d) neither parent?

There is a \(50-50\) chance that the queen carries the gene for hemophilia. If she is a carrier, then each prince has a \(50-50\) chance of having hemophilia. If the queen has had three princes without the disease, what is the probability that the queen is a carrier? If there is a fourth prince, what is the probability that he will have hemophilia?

On the morning of September \(30,1982,\) the won lost records of the three leading baseball teams in the Western Division of the National League were as follows: $$\begin{array}{lrr} \hline \text { Team } & \text { Won } & \text { Lost } \\ \hline \text { Atlanta Braves } & 87 & 72 \\ \text { San Francisco Giants } & 86 & 73 \\ \text { Los Angeles Dodgers } & 86 & 73 \\ \hline \end{array}$$ Each team had 3 games remaining. All 3 of the Giants' games were with the Dodgers, and the 3 remaining games of the Braves were against the San Diego Padres. Suppose that the outcomes of all remaining games are independent and each game is equally likely to be won by either participant. For each team, what is the probability that it will win the division title? If two teams tie for first place, they have a playoff game, which each team has an equal chance of winning.

A town council of 7 members contains a steering committee of size \(3 .\) New ideas for legislation go first to the steering committee and then on to the council as a whole if at least 2 of the 3 committee members approve the legislation. Once at the full council, the legislation requires a majority vote (of at least 4 ) to pass. Consider a new piece of legislation, and suppose that each town council member will approve it, independently, with probability \(p .\) What is the probability that a given steering committee member's vote is decisive in the sense that if that person's vote were reversed, then the final fate of the legislation would be reversed? What is the corresponding probability for a given council member not on the steering committee?

Suppose that each child born to a couple is equally likely to be a boy or a girl, independently of the sex distribution of the other children in the family. For a couple having 5 children, compute the probabilities of the following events: (a) All children are of the same sex. (b) The 3 eldest are boys and the others girls. (c) Exactly 3 are boys. (d) The 2 oldest are girls. (e) There is at least 1 girl.

\(A\) and \(B\) alternate rolling a pair of dice, stopping either when \(A\) rolls the sum 9 or when \(B\) rolls the sum \(6 .\) Assuming that \(A\) rolls first, find the probability that the final roll is made by \(A.\)

In a certain village, it is traditional for the eldest son (or the older son in a two-son family) and his wife to be responsible for taking care of his parents as they age. In recent years, however, the women of this village, not wanting that responsibility, have not looked favorably upon marrying an eldest son. (a) If every family in the village has two children, what proportion of all sons are older sons? (b) If every family in the village has three children, what proportion of all sons are eldest sons? Assume that each child is, independently, equally likely to be either a boy or a girl.

Suppose that \(E\) and \(F\) are mutually exclusive events of an experiment. Show that if independent trials of this experiment are performed, then \(E\) will occur before \(F\) with probability \(P(E) /[P(E)+\) \(P(F)].\)

Consider an unending sequence of independent trials, where each trial is equally likely to result in any of the outcomes \(1,2,\) or \(3 .\) Given that outcome 3 is the last of the three outcomes to occur, find the conditional probability that. (a) the first trial results in outcome 1. (b) the first two trials both result in outcome \(1 .\)

\(A\) and \(B\) play a series of games. Each game is independently won by \(A\) with probability \(p\) and by \(B\) with probability \(1-p .\) They stop when the total number of wins of one of the players is two greater than that of the other player. The player with the greater number of total wins is declared the winner of the series. (a) Find the probability that a total of 4 games are played. (b) Find the probability that \(A\) is the winner of the series.

In a certain contest, the players are of equal skill and the probability is \(\frac{1}{2}\) that a specified one of the two contestants will be the victor. In a group of \(2^{n}\) players, the players are paired off against each other at random. The \(2^{n-1}\) winners are again paired off randomly, and so on, until a single winner remains. Consider two specified contestants, \(A\) and \(B\), and define the events \(A_{i}, i \leq n, E\) by \(A_{i}:\) \(A\) plays in exactly \(i\) contests: \(E: \quad A\) and \(B\) never play each other. (a) \(\operatorname{Find} P\left(A_{i}\right), i=1, \ldots, n\) (b) Find \(P(E)\) (c) Let \(P_{n}=P(E) .\) Show that $$ P_{n}=\frac{1}{2^{n}-1}+\frac{2^{n}-2}{2^{n}-1}\left(\frac{1}{2}\right)^{2} P_{n-1} $$ and use this formula to check the answer you obtained in part (b). Hint: Find \(P(E)\) by conditioning on which of the events \(A_{i}, i=1, \ldots, n\) occur. In simplifying your answer, use the algebraic identity $$ \sum_{i=1}^{n-1} i x^{i-1}=\frac{1-n x^{n-1}+(n-1) x^{n}}{(1-x)^{2}} $$ For another approach to solving this problem, note that there are a total of \(2^{n}-1\) games played. (d) Explain why \(2^{n}-1\) games are played. Number these games, and let \(B_{i}\) denote the event that \(A\) and \(B\) play each other in game \(i, i=1, \ldots, 2^{n}-1\) (e) What is \(P\left(\bar{B}_{i}\right) ?\) (f) Use part (e) to find \(P(E).\)

An investor owns shares in a stock whose present value is \(25 .\) She has decided that she must sell her stock if it goes either down to 10 or up to \(40 .\) If each change of price is either up 1 point with probability .55 or down 1 point with probability \(.45,\) and the successive changes are independent, what is the probability that the investor retires a winner?

\(A\) and \(B\) flip coins. \(A\) starts and continues flipping until a tail occurs, at which point \(B\) starts flipping and continues until there is a tail. Then \(A\) takes over, and so on. Let \(P_{1}\) be the probability of the coin's landing on heads when \(A\) flips and \(P_{2}\) when \(B\) flips. The winner of the game is the first one to get (a) 2 heads in a row; (b) a total of 2 heads; (c) 3 heads in a row; (d) a total of 3 heads. In each case, find the probability that \(A\) wins.

Die \(A\) has 4 red and 2 white faces, whereas die \(B\) has 2 red and 4 white faces. A fair coin is flipped once. If it lands on heads, the game continues with dic \(A ;\) if it lands on tails, then dic \(B\) is to be used. (a) Show that the probability of red at any throw is \(\frac{1}{2}.\) (b) If the first two throws result in red, what is the probability of red at the third throw? (c) If red turns up at the first two throws, what is the probability that it is die \(A\) that is being used?

An urn contains 12 balls, of which 4 are white. Three players \(-A, B,\) and \(C-\) successively draw from the urn, \(A\) first, then \(B\), then \(C\), then \(\bar{A}\), and so on. The winner is the first one to draw a white ball. Find the probability of winning for each player if (a) each ball is replaced after it is drawn; (b) the balls that are withdrawn are not replaced.

Let \(S=\\{1,2, \ldots, n\\}\) and suppose that \(A\) and \(B\) are, independently, equally likely to be any of the \(2^{n}\) subsets (including the null set and \(S\) itself) of \(S\) (a) Show that $$ P\\{A \subset B\\}=\left(\frac{3}{4}\right)^{n} $$ Hint: Let \(N(B)\) denote the number of elements in \(B\). Use \(P\\{A \subset B\\}=\sum_{i=0}^{n} P\\{A \subset B | N(B)=i\\} P\\{N(B)=i\\}\) Show that \(P\\{A B=\varnothing\\}=\left(\frac{3}{4}\right)^{n}.\)

A person tried by a 3 -judge panel is declared guilty if at least 2 judges cast votes of guilty. 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 .2. If 70 percent of defendants are guilty, compute the conditional probability that judge number 3 votes guilty given that (a) judges 1 and 2 vote guilty; (b) judges 1 and 2 cast 1 guilty and 1 not guilty vote; (c) judges 1 and 2 both cast not guilty votes. Let \(E_{i}, i=1,2,3\) denote the event that judge \(i\) casts a guilty vote. Are these events independent. Are they conditionally independent? Explain.

Suppose that \(n\) independent trials, each of which results in any of the outcomes \(0,1,\) or \(2,\) with respective probabilities \(p_{0}, p_{1},\) and \(p_{2}, \sum_{i=0}^{2} p_{i}=1\) are performed. Find the probability that outcomes 1 and 2 both occur at least once.