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An experiment involves tossing a single die. These are some events: A: Observe a 2 \(B:\) Observe an even number \(C:\) Observe a number greater than 2 \(D:\) Observe both \(A\) and \(B\) \(E:\) Observe \(A\) or \(B\) or both \(F:\) Observe both \(A\) and \(C\) a. List the simple events in the sample space. b. List the simple events in each of the events \(A\) through \(F\) c. What probabilities should you assign to the simple events? d. Calculate the probabilities of the six events \(A\) through \(F\) by adding the appropriate simple-event probabilities.

A sample space \(S\) consists of five simple events with these probabilities: $$\begin{array}{c}P\left(E_{1}\right)=P\left(E_{2}\right)=.15 \quad P\left(E_{3}\right)=.4\\\P\left(E_{4}\right)=2 P\left(E_{5}\right)\end{array}$$ a. Find the probabilities for simple events \(E_{4}\) and \(E_{5}\). b. Find the probabilities for these two events: $$A=\left\\{E_{1}, E_{3}, E_{4}\right\\}$$ $$B=\left\\{E_{2}, E_{3}\right\\}$$ c. List the simple events that are either in event \(A\) or event \(B\) or both. d. List the simple events that are in both event \(A\) and event \(B\).

A jar contains four coins: a nickel, a dime, a quarter, and a half-dollar. Three coins are randomly selected from the jar. a. List the simple events in \(S\). b. What is the probability that the selection will contain the half-dollar? c. What is the probability that the total amount drawn will equal \(60 \phi\) or more?

A survey classified a large number of adults according to whether they were judged to need eyeglasses to correct their reading vision and whether they used eyeglasses when reading. The proportions falling into the four categories are shown in the table. (Note that a small proportion, .02, of adults used eyeglasses when in fact they were judged not to need them.) $$\begin{array}{lcc} & \begin{array}{l}\text { Used Eyeglasses } \\\\\text { for Reading }\end{array} \\\\\hline \text { Judged to Need } & & \\ \text { Eyeglasses } & \text { Yes } & \text { No } \\\\\hline \text { Yes } & .44 & .14 \\\\\text { No } & .02 & .40\end{array}$$ If a single adult is selected from this large group, find the probability of each event: a. The adult is judged to need eyeglasses. b. The adult needs eyeglasses for reading but does not use them. c. The adult uses eyeglasses for reading whether he or she needs them or not.

The game of roulette uses a wheel containing 38 pockets. Thirty-six pockets are numbered \(1,2, \ldots, 36,\) and the remaining two are marked 0 and \(00 .\) The wheel is spun, and a pocket is identified as the "winner." Assume that the observance of any one pocket is just as likely as any other. a. Identify the simple events in a single spin of the roulette wheel b. Assign probabilities to the simple events. c. Let \(A\) be the event that you observe either a 0 or a 00\. List the simple events in the event \(A\) and find \(P(A)\) d. Suppose you placed bets on the numbers 1 through 18. What is the probability that one of your numbers is the winner?

Three people are randomly selected from voter registration and driving records to report for jury duty. The gender of each person is noted by the county clerk. a. Define the experiment. b. List the simple events in \(S\). c. If each person is just as likely to be a man as a woman, what probability do you assign to each simple event? d. What is the probability that only one of the three is a man? e. What is the probability that all three are women?

Four equally qualified runners, John, Bill, Ed, and Dave, run a 100-meter sprint, and the order of finish is recorded. a. How many simple events are in the sample space? b. If the runners are equally qualified, what probability should you assign to each simple event? c. What is the probability that Dave wins the race? d. What is the probability that Dave wins and John places second? e. What is the probability that Ed finishes last?

You have two groups of distinctly different items, 10 in the first group and 8 in the second. If you select one item from each group, how many different pairs can you form?

Permutations Evaluate the following permutations. a. \(P_{3}^{5}\) b. \(P_{9}^{10}\) c. \(P_{6}^{6}\) d. \(P_{1}^{20}\)

Five cards are selected from a 52 -card deck for a poker hand. a. How many simple events are in the sample space? b. A royal flush is a hand that contains the \(\mathrm{A}, \mathrm{K}, \mathrm{Q}, \mathrm{J},\) and \(10,\) all in the same suit. How many ways are there to get a royal flush? c. What is the probability of being dealt a royal flush?