91Ó°ÊÓ

Consider two ums, each containing both white and black balls. The probabilities of drawing white balls from the first and second urns are, respectively, \(p\) and \(p^{\prime}\). Balls are sequentially selected with replacement as follows: With probability \(\alpha\) a ball is initially chosen from the first urn, and with probability \(1-\alpha\) it is chosen from the second urn. The subsequent selections are then made according to the rule that whenever a white ball is drawn (and replaced), the next ball is drawn from the same urn; but when a black ball is drawn, the next ball is taken from the other urn. Let \(\alpha_{n}\) denote the probability that the \(n\)th ball is chosen from the first urn. Show that $$ \alpha_{n+1}=\alpha_{n}\left(p+p^{\prime}-1\right)+1-p^{\prime} \quad n \geq 1 $$ and use this to prove that $$ \alpha_{n}=\frac{1-p^{\prime}}{2-p-p^{\prime}}+\left(\alpha-\frac{1-p^{\prime}}{2-p-p^{\prime}}\right)\left(p+p^{\prime}-1\right)^{n-1} $$ Let \(P_{n}\) denote the probability that the \(n\)th ball selected is white. Find \(P_{n}\). Also compute \(\lim _{n \rightarrow \infty} \alpha_{n}\) and \(\lim _{n \rightarrow \infty} P_{n}\).

Rank the following from most likely to least likely to occur. 1\. A fair coin lands on heads. 2\. Three independent trials, each of which is a success with probability .8, all result in successes. 3\. Seven independent trials, each of which is a success with probability .9, all results in successes.

The Ballot Problem. In an election, candidate \(A\) receives \(n\) votes and candidate \(B\) receives \(m\) votes, where \(n>m\). Assuming that all of the \((n+m) ! / n ! m !\) orderings of the votes are equally likely, let \(P_{n, m}\) denote the probability that \(A\) is always ahead in the counting of the votes. (a) Compute \(P_{2,1}, P_{3,1}, P_{3,2}, P_{4,1}, P_{4,2}, P_{4,3}\). (b) Find \(P_{n, 1}, P_{n, 2}\) (c) Based on your results in parts (a) and (b), conjecture the value of \(P_{n, m}\). (d) Derive a recursion for \(P_{n, m}\) in terms of \(P_{n-1, m}\) and \(P_{n, m-1}\) by conditioning on who receives the last vote. (e) Use part (d) to verify your conjecture in part (c) by an induction proof on \(n+m\).

An urn initially contains 5 white and 7 black balls. Each time a ball is selected, its color is noted and it is replaced in the urn along with 2 other balls of the same color. Compute the probability that (a) the first 2 balls selected are black and the next 2 white; (b) of the first 4 balls selected, exactly 2 are black.

As a simplified model for weather forecasting, suppose that the weather (either wet or dry) tomorrow will be the same as the weather today with probability \(p\). If the weather is dry on January 1, show that \(P_{n}\), the probability that it will be dry \(n\) days later, satisfies $$ \begin{aligned} &P_{n}=(2 p-1) P_{n-1}+(1-p) \quad n \geq 1 \\ &P_{0}=1 \end{aligned} $$ Prove that $$ P_{n}=\frac{1}{2}+\frac{1}{2}(2 p-1)^{n} \quad n \geq 0 $$

One probability class of 30 students contains 15 that are good, 10 that are average, and 5 that are of poor quality. A second probability class, also of 30 students, contains 5 that are good, 10 that are fair, and 15 that are poor. You (the expert) are aware of these numbers, but you have no idea which class is which. If you examine one student selected at random from each class and find that the student from class \(A\) is a fair student whereas the student from class \(B\) is a poor student, what is the probability that class \(A\) is the superior class?

Consider a sample of size 3 drawn in the following manner: We start with an urn containing 5 white and 7 red balls. At each stage a ball is drawn and its color is noted. The ball is then returned to the um along with an additional ball of the same color. Find the probability that the sample will contain exactly (a) 0 white balls; (b) 1 white ball; (c) 3 white balls; (d) 2 white balls.

There are 3 coins in a box. One is a two-headed coin; another is a fair coin; and the third is a biased coin that comes up heads 75 percent of the time. When one of the 3 coins is selected at random and flipped, it shows heads. What is the probability that it was the two-headed coin?

Consider the following game. A deck of cards is shuffled and its cards are turned face up one at a time. At any time you can elect to say "next," and if the next card is the ace of spades, then you win, and if not, then you lose. Of course, if the ace of spades appears before you say "next," then you lose. Also, if there is only one card remaining, the ace of spades hasn't yet appeared, and you have never said "next," then you are a winner (since you will say "next"). Argue that no matter what strategy you employ for deciding when to say "next," your probability of winning is \(\frac{1}{52}\).

Suppose that there was a cancer diagnostic test that was 95 percent accurate both on those that do and those that do not have the disease. If \(.4\) percent of the population have cancer, compute the probability that a tested person has cancer, given that his or her test result indicates so.