Overcards: Part I

Intuition is a much discussed - though somewhat elusive - topic. Many people when queried will claim they know what intuition is, but find it difficult to give a clear explanation if pressed. Mathematicians frequently talk about people possessing good ``mathematical intuition'', or talk about a problem having an ``unintuitive'' answer. What do they mean by these expressions? In the first case, I believe we are referring to someone who, when confronted with a research problem, has a feeling about the form the answer should take when the answer is unknown, or has a feeling about the route to follow in order to prove a certain answer is correct when the answer is ``known''. In the second case, I believe we are referring to an answer which is surprisingly different from the answer we would have guessed after a cursory inspection of the problem. Let's look at an example of the latter from a recent poker session in which I participated.

I was chatting quietly with my neighbor on the left when my neighbor on the right had his pocket queens lose to a pair of kings - one of which had appeared on the flop. Upon seeing the loser's unhappiness, the player on my left quietly said to me, ``Doesn't he know that almost half the time an overcard to queens will come in the flop?''

The losing player overheard and responded, ``That can't be right! There are only two ranks bigger than a queen.''

There it was! The losing player's intuitive answer is the likelihood of an overcard to queens being small because there are 10 ranks smaller than queen and only two ranks larger than queen. Let's go beyond this cursory examination and determine the exact probability of an overcard.

We take the viewpoint of the player with pocket queens. There are 50 cards he cannot see. We are choosing three cards from 50 for the flop, so there are C(50,3) = 50!/3!47! = 19,600 flops. How many of the flops contain at least one ace or king? The easiest way to determine this is to count the number of flops containing none of them and subtract it from 19,600. Of the 50 unseen cards, 42 remain if we disallow aces and kings. Thus, there are C(42,3)=11,480 flops containing neither an ace nor a king. Hence, there are 8,120 flops containing at least one ace or king. Then the probability of an overcard coming in the flop is 8,120/19,600 = 0.414. This undoubtedly would surprise the losing player and he would interpret it as an unintuitive result.

To calculate the probability of an overcard in the board, we perform similar calculations. There are C(50,5)=2,118,760 boards and there are C(42,5)=850,668 boards without an overcard. The number of boards with at least one overcard is 1,268,092. This implies the probability of at least one overcard in the board is 0.599.

We have seen that 60 percentof the time an overcard to queens will appear in the board. We calculate in the same way the probabilities of overcards to any pocket pair. The probabilities are given in the table below. The column headed ``flop'' is the probability of at least one overcard appearing in the flop, and the column headed ``board'' is the probability of at least one overcard appearing in the board. Readers may find many of these values unintuitive.