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# Expected Value

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I draw a card from a standard 52-card deck. If I draw an Ace, I win 1 dollar. If I draw a 2 through 10, I win a number of dollars equal to the value of the card. If I draw a face card (Jack, Queen, or King), I win 20 dollars. If I draw a club, my winnings are doubled, and if I draw a spade, my winnings are tripled. (For example, if I draw the 8club, then I win 16 dollars.) What would be a fair price to pay to play the game? Express your answer as a dollar value rounded to the nearest cent.

Feb 5, 2023

#1
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The expected value of a single draw is equal to the sum of the probability of each outcome multiplied by the value of the winnings for that outcome.

Let's calculate the expected value:

Ace: 4/52 * 1 = 0.153846

2 through 10: 40/52 * (2 + 3 + ... + 10) / 10 = 1.53846

Face card: 4/52 * 20 = 1.53846

Club: 13/52 * (1 + 2 + ... + 20) / 10 = 2.692308

Spade: 13/52 * (1 * 3 + 2 * 3 + ... + 20 * 3) / 10 = 4.038462

Expected value = 0.153846 + 1.53846 + 1.53846 + 2.692308 + 4.038462, which rounds to 9.19.

Feb 5, 2023
#2
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By my calculations, the answer is

1/4*2*(3/13*1 + 1/13*(2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10) + 9/13(20)) = 9.12 dollars

Feb 5, 2023