Charlie Tuna puts two decks of well shuffled playing cards...

Charlie Tuna puts two decks of well shuffled playing cards side by side in front of you. He begins by simultaneously turning over a card in each deck. He does this, over and over, until all cards have been turned over in pairs. If, on any turn, Charlie hits the same card in both decks you win a dollar. If he has no matches you lose a dollar. (You win only one dollar if he has multiple matches.) What is the probability you will win?

Key Concepts

Complementary Probability

Complementary probability is the method of finding the probability of an event by subtracting the probability of its complement from one. In this case, it involves computing the probability of getting no matches (using the concept of derangements) and then subtracting that value from one to determine the chance of winning (having at least one match). This approach simplifies solving problems where directly computing the desired probability is complex.

Derangements

A derangement is a permutation with no fixed points; that is, no element appears in its original position. In matching problems, calculating the number of derangements helps in determining the likelihood that no match occurs between the two decks. This concept is key for finding the probability of failure, which can then be used to find the probability of success by taking its complement.

Random Permutations

A random permutation involves arranging a set of objects in a completely random order, with each possible arrangement being equally likely. In problems like this, one deck of cards can be seen as a random permutation of another, where the positions correspond to pairs of cards. This concept is fundamental in understanding the probability structure behind matching problems.

Fixed Points in Permutations

Fixed points occur in a permutation when an element’s position remains unchanged relative to a reference order. In the context of this problem, a fixed point represents a situation where the card from one deck is identical to the card from the other deck in the same position. Recognizing fixed points helps in framing the problem as one involving coincidences or matches.

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