Exam 9: Counting, Probability Distributions, and Further Topics in Probability

Solve the problem. -A ring contains 8 keys: 1 red, 1 blue, and 6 gold. If the keys are arranged at random on the ring, find the probability that the red is next to the blue.

Find the expected value for the random variable. -

Evaluate the expression. - ${ }_{38} \mathrm{C}_{1}$

Solve the problem. -Five cards are drawn at random from an ordinary deck of 52 cards. In how many ways is it possible to draw all black cards?

Prepare a probability distribution for the experiment. Let $x$ represent the random variable, and let $P$ represent theprobability. -T hree coins are tossed, and the number of tails is noted.

In a certain college, $33 \%$ of the physics majors belong to ethnic minorities. Find the probability of the event from arandom sample of 10 students who are physics majors. -Two or less belong to an ethnic minority.

Solve the problem. -Find the expected number of boys in a family of 4 children.

Find the equilibrium vector for the transition matrix. - $\left[\begin{array}{ll}0.3 & 0.7 \\ 0.6 & 0.4\end{array}\right]$

Find the expected value of the optimal strategy. -A backpacker is planning a three-day trip in the backcountry with a friend. She wishes to keep the weight of her pack to a minimum. She has decided to bring either rock climbing shoes or an extra fleece jacket but not both. The fleece jacket will be useful if the weather is cold. The rock climbing shoes will be useful if the weather is warm. In trying to decide which of these two items to bring, the backpacker decides on the utilities shown in the following payoff matrix: According to the weather forecaster, the probability of cold weather is 0.4 . The backpacker selects the item that will maximize the expected utility. What is the expected utility in this case?

Solve the problem. -At the first tri-city meeting, there were 8 people from town $A, 7$ people from town $B$ , and 5 people from town C. If the council consists of 5 people, find the probability of 3 from town $A$ and 2 from town B.

A bag contains 6 cherry, 3 orange, and 2 lemon candies. You reach in and take 3 pieces of candy at random. Find the probability. -2 cherry, 1 lemon

Use the multiplication principle to solve the problem. -A restaurant offers 6 possible appetizers, 10 possible main courses, and 4 possible desserts. How many different meals are possible at this restaurant? (Two meals are considered different unless all three courses are the same).

A die is rolled 20 times and the number of twos that come up is tallied. Find the probability of getting the given result. -Less than four twos

Solve the problem. -In a certain lottery, 4 numbers between 1 and 10 inclusive are drawn. These are the winning numbers. How many different selections are possible? Assume that the order in which the numbers are drawn is not important.

A die is rolled five times and the number of fours that come up is tallied. Find the probability of getting the given result. -Exactly four fours

Solve the problem. -A musician plans to perform 10 selections. In how many ways can she arrange the musical selections?

Find the equilibrium vector for the transition matrix. - $\left[\begin{array}{rrr} 0.1 & 0.3 & 0.6 \\0)25 & 0.35 & 0.4 \\0)55 & 0.4 & 0.05\end{array}\right]$

Solve the problem. -From a group of 10 women and 15 men, a researcher wants to randomly select 7 women and 7 men for a study. In how many ways can the study group be selected?

Use the payoff matrix to determine the best strategy. -A college is holding a rock and roll concert to be held in the stadium and has sold out all tickets. If it rains, the show will have to be moved to the gym which has a smaller seating capacity. The college must decide in advance whether to set up seats and the stage in the gym, or in the stadium, or both, just in case. The net profit in each case is given in the payoff matrix below: What strategy should the organizer choose if she is a pessimist?

A die is rolled five times and the number of fours that come up is tallied. Find the probability of getting the given result. -Exactly three fours