91Ó°ÊÓ

In a certain population, the probability a woman lives to at least seventy years is 0.70 and is 0.55 that she will live to at least eighty years. If a woman is seventy years old, what is the conditional probability she will survive to eighty years? Note that if \(A \subset B\) then \(P(A B)=P(A)\).

Two fair dice are rolled. a. What is the (conditional) probability that one turns up two spots, given they show different numbers? b. What is the (conditional) probability that the first turns up six, given that the sum is \(k,\) for each \(k\) from two through \(12 ?\) c. What is the (conditional) probability that at least one turns up six, given that the sum is \(k\), for each \(k\) from two through \(12 ?\)

A shipment of 1000 electronic units is received. There is an equally likely probability that there are \(0,1,2,\) or 3 defective units in the lot. If one is selected at random and found to be good, what is the probability of no defective units in the lot?

Polya's urn scheme for a contagious disease. An urn contains initially \(b\) black balls and \(r\) red balls \((r+b=n) .\) A ball is drawn on an equally likely basis from among those in the urn, then replaced along with \(c\) additional balls of the same color. The process is repeated. There are \(n\) balls on the first choice, \(n+c\) balls on the second choice, etc. Let \(B_{k}\) be the event of a black ball on the \(k\) th draw and \(R_{k}\) be the event of a red ball on the \(k\) th draw. Determine a. \(P\left(B_{2} \mid R_{1}\right)\) b. \(P\left(B_{1} B_{2}\right)\) c. \(P\left(R_{2}\right)\) d. \(P\left(B_{1} \mid R_{2}\right)\).