# 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.