What are the odds of AA against AJ three hands in a row in texas hold 'em? |ShkyBoy

Answer

This is kind of a tricky question in that it involves a few assumptions which aren't strictly math based. First off we'll assume it is a full ring game with nine players. This is important, since what we are assuming is that any of those players could have the AA and AJ. Second we'll assume that preflop action is such that both people see the flop. Then we'll ignore any further action, whether all-in or further betting.

So we now need to determine what the chance is that one player has AA and any other player has AJ. The chance that a given player has AA is simply the number of possible aces divided by the number of total pockets. Aces are (4c2) = 6 and total pockets is (52c2) = 1326, that gives us 6 / 1326 ~= 0.45%. For AJ there are 2 fewer cards in the deck, and 2 fewer aces. They have 8 ways to form AJ (2 aces * 4 jacks) from a total of (50c2) = 1225 total hands. Combined that gives us (6/1326) * (8/1225) ~= 0.003%. A very low number.

But wait, we haven't answered the correct question. That number is actually the chance that the first player gets aces, and the second player ace-jack. We don't care which players get them, so long as two players do. Trying to directly calculate "any two players" is difficult. So instead we'll keep our answer to explicitly mean the first and second player, and then simply permute the players to those positions. For this we say (9p2) = 72, in this case you could also simply say 9 * 8 = 72, the number of ordered two player sets. We now multiply this by the answer we got before to get our chance (6/1326) * (8/1225) * 72 ~= 0.21%. Or about once every 470 hands.

Of course, you asked what is the chance of this happening three times in a row. You can get this simply be raising 0.21% to the power of 3, a very low number. About 1 in 103 million. Though you should actually ignore the first occurrence, since the result was only interesting when it happened two more times. That is, the first time it happens is simply a matter of waiting some 470 hands, and then what is the chance that it happens twice more. Here the number is around 1 in 220 thousand.

Refer to combinations.