91Ó°ÊÓ

Consider an urn containing 12 balls, of which 8 are white. A sample of size 4 is to be drawn with replacement (without replacement). What is the conditional probability (in each case) that the first and third balls drawn will be white given that the sample drawn contains exactly 3 white balls?

The king comes from a family of 2 children. What is the probability that the other child is his sister??????? probability that the other child his? ??????? the ??

Suppose that an ordinary deck of 52 cards is shuffled and the cards are then turned over one at a time until the first ace appears. Given that the first ace is the 20 th card to appear, what is the conditional probability that the card following it is the (a) ace of spades? (b) two of clubs?

On rainy days, Joe is late to work with probability \(.3 ;\) on nonrainy days, he is late with probability . \(1 .\) With probability.7, it will rain tomorrow. (a) Find the probability that Joe is early tomorrow. (b) Given that Joe was early, what is the conditional probability that it rained?

In Example \(3 \mathrm{f},\) suppose that the new evidence is subject to different possible interpretations and in fact shows only that it is 90 percent likely that the criminal possesses the characteristic in question. In this case, how likely would it be that the suspect is guilty (assuming, as before, that he has the characteristic)?

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?

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?

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?

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 \(1 .\)

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 \(P(E).\)