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I have 6 cards on the table, Ace through 6. They are randomly placed face down in a row. I try to guess the correct order of the cards. It is possible to have zero correct guesses, or one correct, etc. I would like to know the correct distribution of the probabilities for the possible results of the guess.

user306996
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    I think the distribution does not have a standard name. For general facts about your problem, you may want to look at derangements (Wikipedia). – André Nicolas Jan 21 '16 at 22:17
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The probability of zero cards correct is 5!/6! = 1/6 The probability of one card correct is 6*4!/6! = 1/5 The probability of two correct cards is 6c2*3!/6! = 1/8 The probability of three correct cards is 6c3*2!/6! = 1/18 The probability of four correct cards is 6c4*1!/6! = 1/48 The probability of five correct cards is not possible as the 6th card is also correct The probability of six correct cards is 1/6! = 1/720

These probabilities should sum to one, but they do not come close.

user306996
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  • For the probability of $0$ correct, look up $D(6)$, and divide by $6!$. They may use other notation, like $!6$. My memory is poor, but I sort of recall that $D(6)=265$. – André Nicolas Jan 21 '16 at 22:37
  • For the number with $1$ correct, it could be any of the six. The rest have to be wrong so we get $6D(5)$. Then as usual divide by $6!$ for the probability. It would be much better if your answer here were part of the problem statement. – André Nicolas Jan 21 '16 at 22:41