91Ó°ÊÓ

A box contains 3 marbles, 1 red, 1 green, and 1 blue. Consider an experiment that consists of taking 1 marble from the box, then replacing it in the box and drawing a second marble from the box. Describe the sample space. Repeat when the second marble is drawn without first replacing the first marble.

A customer visiting the suit department of a certain store will purchase a suit with probability .22, a shirt with probability .30, and a tie with probability 28. The customer will purchase both a suit and a shirt with probability .11, both a suit and a tie with probability .14, and both a shirt and a tie with probability ,10. A customer will purchase all 3 items with probability \(.06 .\) What is the probability that a customer purchases (a) none of these items: (b) exactly 1 of these items?

A die is rolled continually until a 6 appears, at which point the experiment stops. What is the sample space of this experiment? Let \(E_{n}\) denote the event that \(n\) rolls are necessary to complete the experiment. What points of the sample space are contained in \(E_{n}\) ? What is \(\left(\bigcup_{1}^{x} E_{n}\right)^{c} ?\)

Two dice are thrown. Let \(E\) be the event that the sum of the dice is odd; let \(F\) be the event that at least one of the dice lands on 1 ; and let \(G\) be the event that the sum is 5 . Describe the events \(E F, E \cup F, F G, E F^{c}\), and \(E F G\).

Let \(A\) denote the event that the midtown temperature in Los Angeles is \(70^{\circ} \mathrm{F}\), and let \(B\) denote the event that the midtown temperature in New York is \(70^{\circ} \mathrm{F}\). Also, let \(C\) denote the event that the maximum of the midtown temperatures in New York and in Los Angeles is \(70^{\circ} \mathrm{F}\). If \(P(A)=.3, P(B)=.4\), and \(P(C)=.2\), find the probability that the minimum of the two midtown temperatures is \(70^{\circ} \mathrm{F}\).

For any sequence of events \(E_{1}, E_{2}, \ldots\), define a new sequence \(F_{1}, F_{2}, \ldots\) of disjoint events (that is, events such that \(F_{i} F_{j}=\varnothing\) whenever \(i \neq j\) ) such that for all \(n \geq 1\), $$ \bigcup_{1}^{n} F_{i}=\bigcup_{1}^{n} E_{i} $$

An ordinary deck of 52 cards is shuffied. What is the probability that the top four cards have (a) different denominations; (b) different suits?

In a state lottery, a player must choose 8 of the numbers from 1 to 40 . The lottery commission then performs an experiment that selects 8 of these 40 . numbers. Assuming that the choice of the lottery commission is equally likely to be any of the \(\left(\begin{array}{c}40 \\ 8\end{array}\right)\) combinations, what is the probability that a player has (a) all 8 of the numbers selected; (b) 7 of the numbers selected; (c) at least 6 of the numbers selected?

Find the simplest expression for the following events: (a) \((E \cup F)\left(E \cup F^{c}\right)\) (b) \((E \cup F)\left(E^{c} \cup F\right)\left(E \cup F^{c}\right)\); (c) \((E \cup F)(F \cup G)\).

From a group of 3 freshmen, 4 sophomores, 4 juniors, and 3 seniors a committee of size 4 is randomly selected. Find the probability that the committee will consist of (a) 1 from each class; (b) 2 sophomores and 2 juniors; (c) only sophomores or juniors.