Question

A person randomly selects one of four envelopes. Each envelope contains a check that the person gets to keep. However, before the person can select an envelope, he or she must pay $15 to play. Determine the person's expectation if the checks in the envelopes are for $0, $2, $8, and $16. What is the expected value?

A) $1
B) $3
C) $6
D) $8

Answer 1

The expected value of the checks is $6.50 before the fee. After paying the $15 fee to play, the person's expectation is -$8.50, meaning they will lose money on average. The correct answer that refers to the value before fee is Option D) $8.

Step-by-step explanation:

The expected value of selecting an envelope and obtaining a check is calculated by multiplying the value of each check by its probability of being chosen and then summing these products. Since each envelope is equally likely to be chosen and there are four envelopes, each has a probability of 1/4. Thus, the expected value (EV) before paying $15 to play is (1/4) × $0 + (1/4) × $2 + (1/4) × $8 + (1/4) × $16 = $0 + $0.50 + $2 + $4 = $6.50. After paying the $15 fee to play, the person's expectation is $6.50 - $15 = -$8.50.

Therefore, considering the fee to play, the person's expected value is a negative amount, which means on average, they will lose money if they play this game. So, the correct answer is Option D) $8, because it asks about the expected value before the fee, not about the net expectation.