91Ó°ÊÓ

In a certain species of rats, black dominates over brown. Suppose that a black rat with two black parents has a brown sibling. (a) What is the probability that this rat is a pure black rat (as opposed to being a hybrid with one black and one brown gene)? (b) Suppose that when the black rat is mated with a brown rat, all 5 of their offspring are black. Now, what is the probability that the rat is a pure black rat?

Three cards are randomly selected, without replacement, from an ordinary deck of 52 playing cards. Compute the conditional probability that the first card selected is a spade, given that the second and third cards are spades.

If \(0 \leq a_{i} \leq 1, i=1,2, \ldots\), show that $$ \sum_{i=1}^{x}\left[a_{i} \prod_{j=1}^{i-1}\left(1-a_{j}\right)\right]+\prod_{i=1}^{x}\left(1-a_{i}\right)=1 $$ HINT: Suppose that an infinite number of coins are to be flipped. Let \(a_{i}\) be the probability that the ith coin lands heads, and consider when the first head occurs.

The probability of getting a head on a single toss of a coin is \(p\). Consider that \(A\) starts and continues to flip the coin until a tail shows up, at which point \(B\) starts flipping. Then \(B\) continues to flip until a tail comes up, at which point \(A\) takes over, and so on. Let \(P_{n, m}\) denote the probability that \(A\) accumulates a total of \(n\) heads before \(B\) accumulates \(m\). Show that $$ P_{n, m}=p P_{n-1, m}+(1-p)\left(1-P_{m, n}\right) $$

Suppose that you are gambling against an infinitely rich adversary and at each stage you either win or lose 1 unit with respective probabilities \(p\) and \(1-p .\) Show that the probability that you eventually go broke is $$ \begin{array}{cl} 1 & \text { if } p \leq \frac{1}{2} \\ (q / p)^{i} & \text { if } p>\frac{1}{2} \end{array} $$ where \(q=1-p\) and where \(i\) is your initial fortune.

Independent trials that result in a success with probability \(p\) are successively performed until a total of \(r\) successes is obtained. Show that the probability that exactly \(n\) trials are required is $$ \left(\begin{array}{l} n-1 \\ r-1 \end{array}\right) p^{r}(1-p)^{n-r} $$ Use this result to solve the problem of the points (Example 4i). HINT. In order for it to take \(n\) trials to obtain \(r\) successes, how many successes must occur in the first \(n-1\) trials?

Independent trials that result in a success with probability \(p\) and a failure with probability \(1-p\) are called Bernoulli trials. Let \(P_{n}\) denote the probability that \(n\) Bernoulli trials result in an even number of successes ( 0 being considered an even number). Show that $$ P_{n}=p\left(1-P_{n-1}\right)+(1-p) P_{n-1} \quad n \geq 1 $$ and use this to prove (by induction) that $$ P_{n}=\frac{1+(1-2 p)^{n}}{2} $$

Fifty-two percent of the students at a certain college are females. Five percent of the students in this college are majoring in computer science. Two percent of the students are women majoring in computer science. If a student is selected at random, find the conditional probability that (a) this student is female, given that the student is majoring in computer science; (b) this student is majoring in computer science, given that the student is female.

Let \(A\) and \(B\) be events having positive probability. State whether each of the following statements is (i) necessarily true, (ii) necessarily false, or (iii) possibly true. (a) If \(A\) and \(B\) are mutually exclusive, then they are independent. (b) If \(A\) and \(B\) are independent, then they are mutually exclusive. (c) \(P(A)=P(B)=.6\), and \(A\) and \(B\) are mutually exclusive. (d) \(P(A)=P(B)=.6\), and \(A\) and \(B\) are independent.

Consider the gambler's ruin problem with the exception that \(A\) and \(B\) agree to play no more than \(n\) games. Let \(P_{n, i}\) denote the probability that \(A\) winds up with all the money when \(A\) starts with \(i\) and \(B\) with \(N-i\). Derive an equation for \(P_{n, i}\) in terms of \(P_{n-1, i+1}\) and \(P_{n-1, i-1}\) and compute \(P_{7,3}\), \(N=5\)