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

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. -Seven or more belong to an ethnic minority.

Decide whether or not the matrix is a transition matrix. - $\left[\begin{array}{ccc}\frac{4}{8} & \frac{1}{8} & \frac{3}{8} \\1 & 0 & 0 \\\frac{4}{9} & 0 & \frac{5}{9}\end{array}\right]$

Evaluate the expression. - ${ }_{12} \mathrm{C}_{3}$

Construct the transition diagram and the transition matrix that represents the data. -Fifty percent of those who call themselves liberal for the last election will vote as liberals in the next election, $15 \%$ will vote as conservatives, and $35 \%$ will vote as independents. $70 \%$ of those who voted as conservatives in the last election will do so in the next election, while $9 \%$ will vote as liberals, and $21 \%$ will vote as independents. $88 \%$ of those who voted as independents in the last election will do so in the next election, $10 \%$ will vote as liberals, and $2 \%$ will vote as conservatives.

Prepare a probability distribution for the experiment. Let $x$ represent the random variable, and let $P$ represent theprobability. -A ball player batting .300 comes to bat 3 times in a game. The number of hits is counted.

Give the probability distribution and sketch the histogram. -At a safety program for hunters, each member of a class of 25 took 5 shots at a target. The number of bullseyes is shown in the table.

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

Find the requested long-range probabilities based on the transition matrix or data given. -Each week a sales manager classifies her sales people as low, medium, or high producers. The probabilities that a worker will move from a given category to any other category are given by the transition matrix below. Find the long-range trend for the proportion of low, medium, and high producers.

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 orange, 1 lemon

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. -No more than 6 belong to an ethnic minority.

Decide whether or not the matrix is a transition matrix. - $\left[\begin{array}{rr}0.7 & 0 \\ 0 & 0.3\end{array}\right]$

Solve the problem. -A classical music concert is to consist of 5 cello pieces, 3 choral works, and 2 pieces for piano. In how many ways can the program be arranged if a piano piece must come first?

Solve the problem. -An insurance company has written 89 policies of $\$ 50,000,450$ of $\$ 25,000$ , and 900 of $\$ 10,000$ on people of age 20. If the probability that a person will die at age 20 is .001, how much can the company expect to pay during the year the policies were written?

Solve the problem. -A bag contains 4 blue, 3 red, and 5 green marbles. Four marbles are drawn at random from the bag. How many different samples are possible which include at least one marble of each color?

Find the expected value of the optimal 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: According to the weather forecaster, the probability of rain is 0.1 . What is the maximum expected profit?

Use the multiplication principle to solve the problem. -Mark can remember only the first 5 digits of his friend's phone number. He also knows that the number has 7 digits and that the last digit is not a 0 . If Mark were to dial all of the possible numbers and if it takes him 16 seconds to try each one, how long would it take to try every possibility?

Solve the problem. -A pollster wants to minimize the effect the order of the questions has on a person's response to a survey. How many different surveys are required to cover all possible arrangements if there are 8 questions on the survey?

Solve the problem. -In a game of musical chairs, 9 children will sit in 8 chairs arranged in a row (one child will not find a chair). In how many ways can 8 of the children find seats?

Find the requested probability. -A child rolls a 6 -sided die 6 times. What is the probability of the child rolling exactly four fives?

Solve the problem. -Three student representatives are to be chosen from a group of five students: Andrew, Brenda, Chad, Dorothy, and Eric. In how many different ways can the representatives be chosen if two must be male and one female?