Probability Question

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Ryand

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Jan 16, 2020
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If I work through a deck of 52 playing cards, counting from 1/Ace to 13/King a total of four times as I turn over cards, what is the probability that I will turn over a card whose number matches my count before I get through the entire deck?

I have a limited understanding of probabilities, so I'm uncertain about how to calculate the union of these possible outcomes or if that's the right calculation for my problem. Thanks if you can help!
 
This is a challenging problem. The probability of a match changes as the game progresses...

If the first turn is card value "x", then a no-match would mean x≠1, but then you know the deck has fewer "x" cards than the others. So the probability for matching the next card is no longer 1/13.

I have run a computer simulation of this over 10,000,000 games and found that only 162,977 resulted in a no-match. This is approx 1.63% of games.

I can't think of an easy way to obtain an exact answer.
 
This is a slightly more difficult version of derangements (more difficult because there are four of each number); that in itself is quite a challenge, not something for a beginner. You are asking for the probability that the cards in the deck are not a derangement; that is, the number of non-derangements over the total number of permutations.

The fact is, it's easy for a beginner to make up a problem that is challenging for non-beginners. So, @Ryand, if you made this up, be aware that it is beyond you (but will be fun to pursue if you choose to learn combinatorics). If you were assigned it, you must be expected to know a lot more than you say you do!

Please let us know why you want an answer, and what you expect from us.
 
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