Question

If you draw a card with a value of four or less from a standard deck of cards, I will pay you $162. If not, you pay me $45. (Aces are considered the highest card in the deck.)
Step 1 of 2 : Find the expected value of the proposition. Round your answer to two decimal places. Losses must be expressed as negative values.

Answer 1

Answer: 2.77

The expected value is positive, so you expect to gain on average $2.77 per draw.

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Explanation:

Define two events

W = winning = getting a card 4 or less

L = losing = getting 5 or higher

P(W) = probability of winning

P(W) = 12/52 since there are 12 cards that are four or less out of 52

P(W) = 3/13 for any suit of 13, there are 3 cards that are four or less

P(L) = 1-P(W)

P(L) = 1-3/13

P(L) = 13/13 - 3/13

P(L) = 10/13

V(W) = net value of winning

V(W) = 162

V(L) = net value of losing

V(L) = -45

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E(X) = expected value

E(X) = P(W)*V(W) + P(L)*V(L)

E(X) = (3/13)*162 + (10/13)*(-45)

E(X) = 36/13

E(X) = 2.76923076923077

E(X) = 2.77 rounding to two decimal places (to the nearest cent)

This game is in favor to the player since we got a positive expected value. A fair game would occur if the expected value was 0.