# Unusual poker hands

If you like to play poker live or online, then you probable heard someone utter a comment like: "Hey, I've just been dealt the identical two cards I had last hand. I wonder what the chances of that are?".

This question cannot easily be answered because it is ill formed. In other word it can be interpreted in a number of different ways as it depends what is meany by identical 2 card. Here we look at different interpretations for two possible scenarios.

The first case we conider is identical cards. The total number of hold'em hands a player can be dealt is C(52,2) = 1,326. Henceforth the chance that you get the identical two cards you jut received in your previous hand is 1 in 1,326, or around 0.075 percent chance.

If you play online for example 4 tables simultaneously, you may play 200 hands per hour, so on average you will have 2 identical hands every 6 and a half hour. This could be once a day for a very player. Here we are interpreting the question to mean, "What are the chances the player is dealt the identical two cards for the next hand?"

Next we show a more interesting interpretation of the question.

Given that a player plays one single hold'em session where he play n hands, what is the likelihood that he gets two identical hands consecutively at least once?

The total sequences of length n of possible hands he could be dealt is 1,326 raised to the n-th power. This follows because there are 1,326 choices for each hand in the sequence.It is now easy to count the number of sequences without two successive hands that are the same. The first hand may be anything; that is, there are 1,326 choices for the first hand. Once the first hand is chosen, the second hand may be any of 1,325. Once the second hand is chosen, the third hand may be any of 1,325.

We continue in this way and see that the number of sequences of length n with no two successive hands the same is 1,326 multiplied by 1,325 raised to the (n-1)-th power. We then divide this by the total number of sequences to get the probability there are no successive identical hands among the n hands. Subtracting this number from 1 gives the probability of back-to-back identical hands being dealt at least once in a session consisting of n hands.

The table below shows such probabilities for a few values of n. Note that even though there are 1,326 different hands, a session of only 200 hands already gives one a reasonable chance of seeing back-to-back identical hands.

 nomber of hands 2 25 100 200 500 1000 percent chance 0.075 1.8 7.2 13.9 31.4 52.9

We can see in the table that if you play about 1000 hands (a long session indeed), then there is around a 50/50 chance to get dealt the same hand twice in a row. Jonathan is a sponsored pro at pokerstars. He plays millions of hands, so he sees all sorts or unusual hands.