Average rank of one card, ace to king as one to thirteen

Question

Draw one card at random from a standard 52-card deck. Score each card by its rank: ace is 1, the number cards run 2 through 10, jack is 11, queen is 12, and king is 13.

What is the average rank of the card you draw?

Accepted Answer

Every suit holds exactly the ranks 1 through 13, one of each. So instead of totaling the whole deck, notice that a random card's rank is just a random pick from 1 to 13, each equally likely (the four suits do not change which ranks are available). The average of the whole numbers 1 through 13 is their middle value. For a run of evenly spaced numbers, the average is simply the midpoint of the first and last: EV = \frac{1 + 13}{2} = \frac{14}{2} = 7. If you prefer to check by summing: 1+2+\cdots+13 = 91, and there are 13 ranks, so 91/13 = 7. Same answer. Because all four suits carry the same 13…

Average rank of one card, ace to king as one to thirteen

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