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

Solve the problem. -There are 10 members on a board of directors. If they must elect a chairperson, a secretary, and a treasurer, how many different slates of candidates are possible?

Use the multiplication principle to solve the problem. -How many different 5 -digit sequences can be formed using the digits $0,1, \ldots, 9$ if repetition of digits is allowed?

Use the multiplication principle to solve the problem. -How many different 4 -letter radio station call letters can be made if repeats are allowed and the first letter must be $\mathrm{K}$ .

Use the payoff matrix to determine the best strategy. -A farmer must decide on which of two pieces of land to grow his crops. The first piece of land has better soil and will yield a larger harvest. However this piece of land is low lying and is coastal and his crops will be destroyed in the event of a storm and flooding. The second piece of land has poorer soil and will yield a smaller harvest but is at a higher elevation and would not be affected by flooding. He estimates that the payoff matrix is as follows: Based on meteorological records from previous years, the farmer estimates the probability of flooding next year to be 0.1 . What is the farmer's best strategy?

Find the probability of the following card hands-from a 52 card deck. In poker, aces are either high or low. A bridgehand is made up of 13 cards. -In poker, four of a kind (4 cards of the same value)

Decide whether or not the transition matrix is regular. - $\left[\begin{array}{rrrr}0.29 & 0 & 0.71 & 0 \\0.38 & 0.23 & 0 & 0.39 \\0.46 & 0 & 0.43 & 0.11 \\0 & 0.32 & 0.45 & 0.23\end{array}\right]$

Use the multiplication principle to solve the problem. -How many different 7 -digit phone numbers are possible if the first digit cannot be a 0 ?

Find the expected value for the random variable. -

Prepare a probability distribution for the experiment. Let $x$ represent the random variable, and let $P$ represent theprobability. -Three balls are drawn from a bag containing 5 red and 3 green balls. The number of green balls is counted.

Find the expected value for the random variable x having this probability function. - =17 =18 =19 =20 =21

Solve the problem. -An insurance company says that at age 50 one must choose to take $\$ 10,000$ at age $60, \$ 30,000$ at 70 , or $\$ 50,000$ at 80 ( $\$ 0$ death benefit). The probability of living from 50 to 60 is .83 , from 50 to 70 , .61, and from 50 to $80,0.47$ . Find the expected value at each age.

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. -Exactly 4 do not belong to an ethnic minority.

Solve the problem. -How many distinguishable permutations of letters are possible using the letters in the word BASEBALL?

Find the expected value for the random variable. -A business bureau gets complaints as shown in the following table. Find the expected number of complaints per day.

Construct the transition diagram and the transition matrix that represents the data. -90 percent of the people in one generation who have a certain physical characteristic will pass that characteristic on to the next generation. 50 percent of the people in one generation who do not have this characteristic will pass it on to the next generation.

Solve the problem. -Suppose you pay $\$ 1.00$ to roll a fair die with the understanding that you will get back $\$ 3.00$ for rolling 6 or 4 . What are your expected winnings?

Solve the problem. -A gumball machine had 4 red and 5 yellow gumballs left. You put in pennies until you empty the machine. How many different sequences of red and yellow gumballs can come out of the machine?

Solve the problem. -What is the probability that at least 2 students in a class of 36 have the same birthday?

Find the expected value for the random variable x having this probability function. - =10 =12 =14 =16 =18

Give the probability distribution and sketch the histogram. -At a supermarket during the 6 p.m. rush, the manager counted the number of customers waiting in each of the 12 checkout lines. The results are shown in the table.