# Expected Value

Use the table above to determine P(X is greater than or equal to 2). Determine the expected value of X.

PROBLEM 2

Two fair dice are tossed. You bet $5 that you will roll "an even sum". If you roll "an even sum" you win $10. Otherwise you lose the $5 bet. What is the expected return on this game?

PROBLEM 3

Two fair dice are tossed. You bet $1 that you will roll "doubles". If you roll "doubles" you win $60. Otherwise you lose the $1 bet. What is the expected return on this game?

PROBLEM 4

A probability distribution has an expected value (mean) of 54 and a standard deviation of 0.7. Use Chebyshev's inequality to find the minimum probability that an outcome is between 40 and 68.

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#### Solution Preview

I've put explanations in with the Word document. See the attached file.

PROBLEM 1

Random variable X 0 1 2 3

P (X=x) .125 .375 .250 .250

Use the table above to determine P(X is greater than or equal to 2). Determine the expected value of X.

So looking at the table, P(X>=2) = P(X=2) + P(X=3)

Since these are the only possible values of X that are equal to or greater than (>=) 2.

The table tells us the probabilities of X=2 and X=3 are each 0.25.

Therefore, P(X>=2) = 0.5

The expected value of a random variable is defined as the sum of

x P(X=x) = (0 x 0.125) + (1 x 0.375) + (2 x 0.25) + (3 x 0.25) = 1.625

PROBLEM 2

Two fair dice are tossed. You bet $5 that you will roll "an even sum". If you roll "an even sum" you win $10. ...

#### Solution Summary

A probability distribution has an expected value (mean) of 54 and a standard deviation of 0.7 are calculated. The Chebyshev's inequality is used to find the minimum probability that an outcome is between 40 and 68.

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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