Chapter 3

A fair die is thrown twice. \(A\) is the event "sum of the throws equals \(4, "\) \(B\) is "at least one of the throws is a \(3 . "\) a. Calculate \(\mathrm{P}(A \mid B)\). b. Are \(A\) and \(B\) independent events?

We draw two cards from a regular deck of \(52 .\) Let \(S_{1}\) be the event "the first one is a spade," and \(S_{2}\) "the second one is a spade." a. Compute \(\mathrm{P}\left(S_{1}\right), \mathrm{P}\left(S_{2} \mid S_{1}\right)\), and \(\mathrm{P}\left(S_{2} \mid S_{1}^{c}\right)\). b. Compute \(\mathrm{P}\left(S_{2}\right)\) by conditioning on whether the first card is a spade.

A ball is drawn at random from an urn containing one red and one white ball. If the white ball is drawn, it is put back into the urn. If the red ball is drawn, it is returned to the urn together with two more red balls. Then a second draw is made. What is the probability a red ball was drawn on both the first and the second draws?

We choose a month of the year, in such a manner that each month has the same probability. Find out whether the following events are independent: a. the events "outcome is an even numbered month" (i.e., February, April, June, etc.) and "outcome is in the first half of the year." b. the events "outcome is an even numbered month" (i.e., February, April, June, etc.) and "outcome is a summer month" (i.e., June, July, August).

Spaceman Spiff's spacecraft has a warning light that is supposed to switch on when the freem blasters are overheated. Let \(W\) be the event "the warning light is switched on" and \(F\) "the freem blasters are overheated." Suppose the probability of freem blaster overheating \(\mathrm{P}(F)\) is \(0.1\), that the light is switched on when they actually are overheated is \(0.99\), and that there is a \(2 \%\) chance that it comes on when nothing is wrong: \(\mathrm{P}\left(W \mid F^{c}\right)=0.02\). a. Determine the probability that the warning light is switched on. b. Determine the conditional probability that the freem blasters are overheated, given that the warning light is on.

A certain grapefruit variety is grown in two regions in southern Spain. Both areas get infested from time to time with parasites that damage the crop. Let \(A\) be the event that region \(R_{1}\) is infested with parasites and \(B\) that region \(R_{2}\) is infested. Suppose \(\mathrm{P}(A)=3 / 4, \mathrm{P}(B)=2 / 5\) and \(\mathrm{P}(A \cup B)=4 / 5\). If the food inspection detects the parasite in a ship carrying grapefruits from \(R_{1}\), what is the probability region \(R_{2}\) is infested as well?

A student takes a multiple-choice exam. Suppose for each question he either knows the answer or gambles and chooses an option at random. Further suppose that if he knows the answer, the probability of a correct answer is 1 , and if he gambles this probability is \(1 / 4\). To pass, students need to answer at least \(60 \%\) of the questions correctly. The student has "studied for a minimal pass," i.e., with probability \(0.6\) he knows the answer to a question. Given that he answers a question correctly, what is the probability that he actually knows the answer?

A breath analyzer, used by the police to test whether drivers exceed the legal limit set for the blood alcohol percentage while driving, is known to satisfy $$ \mathrm{P}(A \mid B)=\mathrm{P}\left(A^{c} \mid B^{c}\right)=p, $$ where \(A\) is the event "breath analyzer indicates that legal limit is exceeded" and \(B\) "driver's blood alcohol percentage exceeds legal limit." On Saturday night about \(5 \%\) of the drivers are known to exceed the limit. a. Describe in words the meaning of \(\mathrm{P}\left(B^{c} \mid A\right)\). b. Determine \(\mathrm{P}\left(B^{c} \mid A\right)\) if \(p=0.95\). c. How big should \(p\) be so that \(\mathrm{P}(B \mid A)=0.9\) ?

The events \(A, B\), and \(C\) satisfy: \(\mathrm{P}(A \mid B \cap C)=1 / 4, \mathrm{P}(B \mid C)=1 / 3\), and \(\mathrm{P}(C)=1 / 2\). Calculate \(\mathrm{P}\left(A^{c} \cap B \cap C\right)\).

Two independent events \(A\) and \(B\) are given, and \(\mathrm{P}(B \mid A \cup B)=2 / 3\). \(\mathrm{P}(A \mid B)=1 / 2\). What is \(\mathrm{P}(B)\) ?

You are diagnosed with an uncommon disease. You know that there only is a \(1 \%\) chance of getting it. Use the letter \(D\) for the event "you have the disease" and \(T\) for "the test says so." It is known that the test is imperfect: \(\mathrm{P}(T \mid D)=0.98\) and \(\mathrm{P}\left(T^{c} \mid D^{c}\right)=0.95\). a. Given that you test positive, what is the probability that you really have the disease? b. You obtain a second opinion: an independent repetition of the test. You test positive again. Given this, what is the probability that you really have the disease?

You and I play a tennis match. It is deuce, which means if you win the next two rallies, you win the game; if I win both rallies, I win the game; if we each win one rally, it is deuce again. Suppose the outcome of a rally is independent of other rallies, and you win a rally with probability \(p\). Let \(W\) be the event "you win the game," \(G\) "the game ends after the next two rallies," and \(D\) "it becomes deuce again." a. Determine \(\mathrm{P}(W \mid G)\). b. Show that \(\mathrm{P}(W)=p^{2}+2 p(1-p) \mathrm{P}(W \mid D)\) and use \(\mathrm{P}(W)=\mathrm{P}(W \mid D)\) (why is this so?) to determine \(\mathrm{P}(W)\). c. Explain why the answers are the same.

Suppose \(A\) and \(B\) are events with \(0<\mathrm{P}(A)<1\) and \(0<\mathrm{P}(B)<1\). a. If \(A\) and \(B\) are disjoint, can they be independent? b. If \(A\) and \(B\) are independent, can they be disjoint? c. If \(A \subset B, \operatorname{can} A\) and \(B\) be independent? d. If \(A\) and \(B\) are independent, can \(A\) and \(A \cup B\) be independent?