Richard Burke

We finished figuring the probability of pocket aces and pocket kings among ten players at a hold’em table in our last column, finishing with the table shown below. The probabilities of each possible way to make two or more pairs with aces and kings sum to 0.00230.

We continue figuring the probability of pocket aces and pocket kings among ten players at a hold’em table. The first table (not shown) shows the known cells and the remaining five cells for which we obtain these probabilities.
 

We return to figuring the probability that two or more players hold pocket pairs at a ten-player table. In a previous column we figured the probability that two players at a ten-player table would have pocket pairs of aces and/or kings, when four aces and kings lie among the top twenty cards in a standard deck. In this column we continue filling out the probability table shown below (not shown):

We turn now to the task of figuring the probability that two or more players hold pocket pairs at a ten player table. We expect this proof to take several columns. We start by figuring the probability that from 0 to 8 of aces and kings lie among the twenty cards dealt to the players.

During the 2008 WSOP Main Event, quad aces lost to a royal flush with both hole cards playing. (You can view the two-minute video at http://www.youtube.com/watch?v=XunAlp2azhA)

In post-production ESPN’s math consultant rummaged around in a place renowned for its lack of sunshine and pulled out a number for the odds. In the show’s airing, ESPN’s Lon McEachern announced that number to the world, 2.7 billion-to-1. Wrong. Nope. Nyet. Non. No way. Not even close—that number misses by two orders of magnitude! Let’s calculate the right odds.

Fred cornered me when I signed up for a few games. He understood the principle behind raisin' and chasin' with modest hands: By raising before the flop you make the pot odds high enough to see the turn card and the river card with remarkably few outs, thus pricing yourself into the hand.

He also understood that if the pot odds for you exceed the cards odds against you, then chasing shows a profit in the long run. He appreciated the charts in Parts 6 and 7 of this series because he could count active opponents better than he could accurately track pot sizes.

You will recall our previous columns on raisin' and chasin' where we raised pre-flop in late position with a modest hand, for example, 6d-5d. In part 6 of this series, we presented a table showing the number of clean outs you need in order to see the turn card. The number of outs was dependent on the number of opponents you could count on to pay to see the turn card also. That table applies to fixed-limit hold 'em where betting limits double on the turn and river. In this column we show the number of outs you need to pay to see the river card.

The table shows the "Rake Effect" at structured, limit hold 'em games where the limits double on the turn, i.e., the pattern, B-B-2B-2B. The table derives from a raise before the flop and one bet after the flop.

You will recall our previous columns where we raised pre-flop in late position with a modest hand, for example, 6d-5d. After the flop, either someone bet or they all checked to us and we bet. In that event the pot would contain three bets from each of the remaining players, plus dead money left by the blinds and by players who folded after seeing the flop, minus the drop.

The table shows the "Rake Effect" at structured, limit hold 'em games where the limits double on the turn, i.e., the pattern, B-B-2B-2B. The table derives from a raise before the flop and one bet after the flop.

You will recall our previous columns where we raised pre-flop in late position with a modest hand, for example, 6d-5d. After the flop, either someone bet or they all checked to us and we bet. In that event the pot would contain three bets from each of the remaining players, plus dead money left by the blinds and by players who folded after seeing the flop, minus the drop.

We interrupt our series of raisin' 'n' chasin' columns to assess the flop. We read in D. R. Sherer's book, No Fold'em Hold'em, that the median flop contains a queen. If true, that explains why our pocket jacks, tens, and nines so often fail to prevail-those pesky overcards kill our chances much too often!

The median flop has just as many flops with higher ranking cards as those with lower-ranking ones. Before looking at our hole cards, C(52,3) describes the total possible number of flops: 22,100. The median flop has 11,050 higher flops and 11,050 lower, by definition.