A correct solution, arrived at by means of numerical simulation, was received from Philippe Fondanaiche.

The second part of the problem is more difficult. You will have less that a 50% chance to have a trio with 13 cards, and more than a 50% chance with 14 cards.

Let p(k) denote the probability that you have a trio after drawing k cards.   Obviously p(2) = 0 while we just saw that p(27) = 1 and it's clear that this probability increases as you draw more cards.  What is the smallest value of k for which 1/2 £ p(k)?

The function p is given by the following equation:

where ëk/2 û means k/2 rounded down to the nearest integer, and  is the number of ways to choose t things from a set of s objects.  A graph of the function appears below.

You can see that 14 cards will be the smallest number for which a probability of a trio is greater than 1/2.

Where does the function come from?  Let's start with the numerator of the fraction; I'll argue it counts the number of ways to choose k cards from a standard deck of 52 cards so that no trio is drawn.  Now, each such choice will contain either no pair, one pair, two pair, and so on, with ë k/2û being the largest number of pairs which can come up.  The summands count the number of ways in which k cards can be chosen with exactly i pairs appearing: first select which of the thirteen faces are to be paired, then select the pair from within the four cards of that face; the other cards must be unpaired, so from the 13 - i remaining faces, choose k - 2i faces, and then one card from each face.  The fraction, then, is the probability of choosing k cards from the deck and failing to get a trio.  Subtracting from one gives the desired probablity.

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Page last updated 1 November 2000.