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World Series. The World Series in baseball is won by the first team to win four games (ignoring the 1903 and 1919–1921 World Series, when it was a best of nine). Thus it takes at least four games and no more than seven games to establish a winner. From the document World Series History on the Baseball Almanac website, as of November 2013, the lengths of the World Series are as given in the following table

a. If X denotes the number of games that it takes to complete a World Series, identify the possible values of the random variable X.

b. Do the first and third columns of the table provide a probability distribution for X? Explain your answer.

c. Historically, what is the most likely number of games it takes to complete a series?

d. Historically, for a randomly chosen series, what is the probability that it ends in five games?

e. Historically, for a randomly chosen series, what is the probability that it ends in five or more games?

f. The data in the table exhibit a statistical oddity. If the two teams in a series are evenly matched and one team is ahead three games to two, either team has the same chance of winning game number six. Thus there should be about an equal number of six-and seven-game series. If the teams are not evenly matched, the series should tend to be shorter, ending in six or fewer games, not seven games. Can you explain why the series tend to last longer than expected?

Archery. An archer shoots an arrow into a square target 6 feet on a side whose center we call the origin. The outcome of this random experiment is the point in the target hit by the arrow. The archer scores 10 points if she hits the bull's eye-a disk of radius 1 foot centered at the origin; she scores 5 points if she hits the ring with inner radius 1 foot and outer radius 2 feet centered at the origin; and she scores 0 points otherwise. Assume that the archer will actually hit the target and is equally likely to hit any portion of the target. For one arrow shot, let S be the score.

(a) Obtain and interpret the probability distribution of the random variable S. (Hint: The area of a square is the square of its side length; the area of a disk is the square of its radius times.)

(b) Use the special addition rule and the probability distribution obtained in part (a) to determine and interpret the probability of each of the following events:

$\{S=5);\{S>0\};\{S≤7);(5<S≤15);\{S<15);\mathrm{and}\{S<0).$

Oklahoma State Officials. Refer to Table 5.1 on page 196.

(a). List the possible samples without replacement of size 3 that can be obtained from the population of five officials. (Hint: There are 10 possible samples.)

If a simple random sample without replacement of three officials is taken from the five officials, determine the probability that

(b). the governor, attorney general, and treasurer are obtained.

(c). the governor and treasurer are included in the sample.

(d). the governor is included in the sample.

In Exercise 5.129 - 5.132, We have provided the probability distributions of the random variables considered in Exercise 5.109 - 5.112 of section 5.4 For each exercise, do the following tasks.

Part (a) Find the mean of the random variable.

Part (b) Obtain the standard deviation of the random variable by using one of the formulas given in definition 5.10.

Dice. The random variable Y is the sum of the dice when two balanced dice are rolled. Its probability distribution is as follows

Part (a) Find and interpret the mean of the random variable.

Part (b) Obtain the standard deviation of the random variable by using one of the formulas given in definition 5.10

Part (c) Construct a probability histogram for the random variable, locate the mean: and show one, two, and three standard deviation intervals.

World series. The World series in baseball is won by the first team to win four games ( ignoring the 1903 and 19919 - 1921 World series, when it was a best of nine). From the document World series history on the baseball Almanac website. as of November 2013, the length of the world series are as given in the following table.

Let X denote the number of games that it take to complete a world series, and let Y denote the number of games that it took to complete a randomly selected world series from among those considered in the table.

Part (a) Determine the mean and standard deviation of the random variable Y. Interpret your resuts.

Part (b) Provide an estimate for the mean and standard deviation of the random variable X. Explain your reasoning

Playing Cards. An ordinary deck of playing cards has 52 cards. There are four suits-spades, hearts, diamonds, and clubs- with 13 cards in each suit. Spades and clubs are black; hearts and diamonds are red. If one of these cards is selected at random, what is the probability that it is

(a). a spade? (b). red? (c). not a club?

Evaluating Investments. An investor plans to put $50.000 in one of four investments. The return on each investment depends on whether next year's economy is strong or weak. The following table summarizes the possible payoffs. in dollars for the four investments.

Let V, W, X and Y denotes the payoffs for the certificate or deposit office complex, land speculation. and technical school, respectively the V, W, X and Y are random variables . assume that nest year's economy has a 40% chance of being strong and a 60% chance of being weak.

Part(a) Find the probability distribution of each random variable V, W, X, and Y

Part (b) Determine the expected value of each random variable.

Part (c) Which investment has the best expected payoffs? the worst?

Part (d) Which investment would you select? Explain

Expected Utility. One method for deciding among various investment involves the concept of expected utility. Economists describe the importance of various levels of wealth by using utility functions. For instance, in most case, a single dollar is more important (has greater utility ) for someone with little wealth than for someone with greater wealth Consider two investments, say investment A and B. Measured in thousand of dollars, suppose that investment A yields 0, 1, and 4 with probability 0.1 and16 with probability 0.5, 0.3 and 0.2 respectively. Let Y denote the yield of an investment. For the two investment, determine and compare.

Part (a) The mean of Y, the expected yield.

Part (b) The mean of $\sqrt{Y}$,the expected utility, using the utility function role="math" localid="1651845051902" $\mathrm{Ï\dots }\left(y\right)=\sqrt{y}$Interpret the utility function $u$

Part (c) The mean of $y^{3/2}$,the expected utility, using the utility function $v\left(y\right)=y^{3/2}$. Interpret the utility function v

Equipment breakdowns: A factory manager collected data on the number of equipment breakdown per day. From those data she derived the probability distribution shown in the following table, Where W denote the number of breakdown on a given day.

Part (a) Determine ${\mathrm{μ}}_{w}and{\mathrm{σ}}_w$, Round your answer for the standard deviation to three decimal places.

Part (b) On average, how many breakdown occur per day?

Part (c) About how many breakdown are expected during a 1 year period assuming 250 work days per year?