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 Number Cards Before Face Cards (Posted on 2019-05-02)
If you turn over a well shuffled deck of cards one card at a time, what is the probability that you'll see at least one of each of the denominations of the numeric cards, ace through ten (1 - 10) before seeing any face card (J, Q, K)?

 Submitted by Charlie Rating: 3.0000 (1 votes) Solution: (Hide) Use the inclusion/exclusion technique: add the individual probabilities of seeing ace, plus of seeing deuce, etc. before seeing a face card. There are of course ten probabilities to add, all of them the same: 4/16 = 1/4; since there are ten of them the total is 10/4 = 5/2. Then subtract out all the pairwise probabilities of seeing an ace or a deuce, an ace or a trey, etc. before the first face card. Each pairwise probability is 8/20 = 2/5, and there are C(10,2) = 45 of them so the total is 90/5 = 18 to be subtracted out. Continue to triples, quadruples, ... , any of all ten, alternately adding and subtracting the totals. ``` -let individual how many sum 1 1/4 10 5/2 2 2/5 45 18 3 12/24 120 60 4 16/28 210 120 5 20/32 252 315/2 6 24/36 210 140 7 28/40 120 84 8 32/44 45 360/11 9 36/48 10 15/2 10 40/52 1 10/13 ``` The probability of getting at least one of each numeric denomination before any face card is therefore 5/2 - 18 + 60 - 120 + 315/2 - 140 + 84 - 360/11 + 15/2 - 10/13 = 1/286 ~= 0.00349650349653097

 Subject Author Date Thanks Steven Lord 2019-05-19 16:55:51 Preliminary solution Steven Lord 2019-05-13 11:22:48 Hint Charlie 2019-05-06 07:51:34 Full solution: part 1 formatting correction FrankM 2019-05-05 11:35:30 Full solution: part 1 FrankM 2019-05-05 11:29:29 A start.... Steven Lord 2019-05-02 17:44:25

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