One card, twenty dollars for a face or its face value

Question

Draw one card at random from a standard 52-card deck. If it is a face card (jack, queen, king) you win a flat $20. Otherwise you win its face value in dollars, with an ace counting as$ 1.

What is the average dollar payout?

Accepted Answer

Expected value is the total payout across all 52 cards divided by 52. This time it is easiest to split the deck into two groups. Face cards. There are 12 face cards (three per suit, four suits), each paying 20: 12 	imes 20 = 240 	ext{ dollars.} Everything else (40 cards). These are the four aces and the number cards 2 through 10. - Aces: four of them at 1 each, adding 4. - Number cards 2 through 10: one suit sums to 2+3+\cdots+10 = 54, and four suits give 4 	imes 54 = 216. The non-face group totals 4 + 216 = 220 dollars. Add the two groups for the deck total: 240 + 220 = 460 	ext{ dollars…

One card, twenty dollars for a face or its face value

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