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Expected Value

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1) In a particular lottery, 3 million tickets are sold each week for 50c apiece. Out of the 3 million tickets, 12,006 are drawn at random and without replacement and awarded prizes: twelve thousand $25 prizes, four $10,000 prizes, one $50,000 prize, and one $200,000 prize. If you purchases a single ticket each week, what is the expected value of this game to you?

2) Let the random variable X be the number of days that a certain patient needs to be in the hospital. Suppose X has the p.m.f. f(x) =(5-x)/10, x = 1,2,3,4. If the patient is to receive $200 from an insurance company for each of the first two days in the hospital and $100 for each day after the first two days, what is the expected payment for the hospitalization?

3) In the gambling game chuck-a-luck, for a $1 bet it is possible to win $1, $2, or $3 with respective probabilities 75/216, 15/216, and 1/216. Let X equal the payoff for this game and find E(X). Note that when a bet is won, the $1 that was bet, in addition to the $1, $2, or $3 that is won, is returned to the bettor.

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The solution contains the determination of the expected value of a random variable in a number of different situations.

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Probabilities and Expected Value

For exercises 1 and 2, determine whether a probability distribution is given. If it is not described, identify the requirements that are not satisfied. If it is described, find its mean and standard deviation.

1. When manufacturing DVDs for Sony, batches of DVDs are randomly selected and the number of defects x is found for each batch.

x P(x)

0 .502
1 .365
2 .098
3 .011
4 .001

2. Air American has a policy of routinely overbooking flights, because past experience shows that some passengers fail to show. The random variable x represents the number of passengers who cannot be boarded because there are more passengers than seats.

x P(x)

0 .805
1 .113
2 .057
3 .009
4 .002

3. Reader's Digest recently ran a sweepstakes in which prizes were listed along with the chances of winning: $1,000,000 (1 chance in 90,000,000), $100,000 (1 chance in 110,000,000), $25,000 (1 chance in 110,000,000), $5000 (1 chance in 36,667,000), and $2500 (1 chance in 27,500,000).

a. Find the expected value of the amount won for one entry.
b. Find the expected value if the cost of entering this sweepstakes is the cost of a postage stamp (37 cents). Is it worth entering this contest?

4. Assume that in a test of a gender-selection technique, a clinical trial results in 12 girls in 14 births. Refer to the table below and find the indicated probabilities.

Probabilities of Girls

x (girls) P(x)

0 .000
1 .001
2 .006
3 .022
4 .061
5 .122
6 .183
7 .209
8 .183
9 .122
10 .061
11 .022
12 .006
13 .001
14 .000

a. Find the probability of exactly 12 girls in 14 births
b. Find the probability of 12 or more girls in 14 births
c. Which probability is relevant for determining whether 12 girls in 14 births is unusually high: the result from part (a) or part (b)?
d. Does 12 girls in 14 births suggest that the gender-selection technique is effective? Why or why not?

For exercise 5, assume that a procedure yields a binomial distribution with a trial repeated n times. Use the binomial probability formula to find the probability of x successes given the probability p of success on a single trial.

5. n = 6, x = 2, p = 0.45

6. The Telektronic Company purchases large shipments of fluorescent bulbs and uses this acceptance sampling plan: Randomly select and test 24 bulbs, then accept the whole batch if there is only one or none that doesn't work. If a particular shipment of thousands of bulbs actually has a 4% rate of defects, what is the probability that this whole shipment will be accepted?

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