Chapter 4

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: $$\begin{array}{l}A=\left\\{E_{1}, E_{3}, E_{4}\right\\} \\\B=\left\\{E_{2}, E_{3}\right\\}\end{array}$$ 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 sample space contains 10 simple events: \(E_{1}\), \(E_{2}, \ldots, E_{10} .\) If \(P\left(E_{1}\right)=3 P\left(E_{2}\right)=.45\) and the remaining simple events are equiprobable, find the probabilities of these remaining simple events.

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?

On the first day of kindergarten, the teacher randomly selects 1 of his 25 students and records the student's gender, as well as whether or not that student had gone to preschool. a. How would you describe the experiment? b. Construct a tree diagram for this experiment. How many simple events are there? c. The table below shows the distribution of the 25 students according to gender and preschool experience. Use the table to assign probabilities to the simple events in part \(b\). $$\begin{array}{lcc} & \text { Male } & \text { Female } \\\\\hline \text { Preschool } & 8 & 9 \\\\\text { No Preschool } & 6 & 2\end{array}$$ d. What is the probability that the randomly selected student is male? What is the probability that the student is a female and did not go to preschool?

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 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?

A food company plans to conduct an experiment to compare its brand of tea with that of two competitors. A single person is hired to taste and rank each of three brands of tea, which are unmarked except for identifying symbols \(A, B\), and \(C\). a. Define the experiment. b. List the simple events in \(S\). c. If the taster has no ability to distinguish a difference in taste among teas, what is the probability that the taster will rank tea type \(A\) as the most desirable? As the least desirable?

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 \(t\) wo 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?