91Ó°ÊÓ

If two fair dice are rolled, what is the conditional probability that the first one lands on 6 given that the sum of the dice is \(i\) ? Compute for all values of \(i\) between 2 and 12 .

The probability that a new car battery functions for over 10,000 miles is .8, the probability that it functions for over 20,000 miles is \(.4\), and the probability that it functions for over 30,000 miles is .1. If a new car battery is still working after 10,000 miles, what is the probability that (a) its total life will exceed 20,000 miles; (b) its additional life will exceed 20,000 miles?

A ball is in any one of \(n\) boxes. It is in the \(i\) th box with probability \(P_{i}\). If the ball is in box \(i\), a search of that box will uncover it with probability \(\alpha_{i}\). Show that the conditional probability that the ball is in box \(j\), given that a search of box \(i\) did not uncover it, is $$ \begin{array}{cl} \frac{P_{j}}{1-\alpha_{i} P_{i}} & \text { if } j \neq i \\ \frac{\left(1-\alpha_{i}\right) P_{i}}{1-\alpha_{i} P_{i}} & \text { if } j=i \end{array} $$

How can 20 balls, 10 white and 10 black, be put into two urns so as to maximize the probability of drawing a white ball if an urn is selected at random and a ball is drawn at random from it?

Urn A contains 2 white balls and 1 black ball, whereas urn \(B\) contains 1 white ball and 5 black balls. A ball is drawn at random from urn \(A\) and placed in urn \(B\). A ball is then drawn from um \(B\). It happens to be white. What is the probability that the ball transferred was white?

What is the probability that at least one of a pair of fair dice lands on 6, given that the sum of the dice is \(i, i=2,3, \ldots, 12 ?\)

An event \(F\) is said to carry negative information about an event \(E\), and we write \(F \searrow E\) if $$ P(E \mid F) \leq P(E) $$ Prove or give counterexamples to the following assertions: (a) If \(F \searrow E\), then \(E \searrow F\). (b) If \(F \searrow E\) and \(E \searrow G\), then \(F \searrow G\). (c) If \(F \searrow E\) and \(G \searrow E\), then \(F G \searrow E\). Repeat parts (a), (b), and (c) when \(\searrow\) is replaced by \(\lambda\), where we say that \(F\) carries positive information about \(E\), written \(F \nearrow E\), when \(P(E \mid F) \geq P(E)\)

A friend randomly chooses two cards, without replacement, from an ordinary deck of 52 playing cards. In each of the following situations, determine the conditional probability that both cards are aces. (a) You ask your friend if one of the cards is the ace of spades and your friend answers in the affirmative. (b) You ask your friend if the first card selected is an ace and your friend answers in the affirmative. (c) You ask your friend if the second card selected is an ace and your friend answers in the affirmative. (d) You ask your friend if either of the cards selected is an ace and your friend answers in the affirmative.

Prove that if \(E_{1}, E_{2}, \ldots, E_{n}\) are independent events, then $$ P\left(E_{1} \cup E_{2} \cup \cdots \cup E_{n}\right)=1-\prod_{i=1}^{n}\left[1-P\left(E_{i}\right)\right] $$

Consider two independent tosses of a fair coin. Let \(A\) be the event that the first toss lands heads, let \(B\) be the event that the second toss lands heads, and let \(C\) be the event that both land on the same side. Show that the events \(A\), \(B, C\) are pairwise independent-that is, \(A\) and \(B\) are independent, \(A\) and \(C\) are independent, and \(B\) and \(C\) are independent-but not independent.