Median Flops

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.

We first consider unpaired flops. Before knowing our hole cards, the total number of unpaired flops equals C(13,3)*43, or 18,304. Dividing that number by the total number of flops, 22,100, shows that 83 percent of the time, or roughly five times out of six, the flop has no duplicated ranks.

To find the median unpaired flop we eliminate ranks. When we eliminate aces, kings, and queens, the number of flops equals C(10,3)*43, or 7680, well below 9152 (half of 18,304). Therefore, the median unpaired flop must have a queen.

Queen what? The number of flops with Q-8-X equals 9024, still less than half of 18,304, so the median flop must also have a nine. Each additional rank adds 64 flops, so the median unpaired flop equals Q-9-3.

If we allow paired or tripped flops, then we add 3796 flops and the median becomes 11,050 flops (half of 21,100).

Because we now allow pairs and trips on the flop, we seek the number, N, for which C(N,3) equals 11,050, or close to it. Our ExcelTM spreadsheet shows 10,660 for COMBIN(41,3) and 11,480 for COMBIN(42,3). Taking eleven or ten cards out of the deck leaves one or two queens and all the lower ranks. Therefore, even including flops with pairs and trips, the median flop still has a queen.

So far, we have yet to see our hole cards. Does knowing our hole cards change the median flop? (Yes, it does!) If we hold two cards higher than a queen, then the median flop can't possibly have either, and it will be lower. Similarly, if we hold two cards lower than a queen, then the median flop will be higher.

Suppose we held pocket queens. The number of possible flops then equals C(50,3), or 19,600. To find the median flop we seek a number, N, for which C(N,3) equals 9,800, or close to it. We find that C(40,3) equals 9880 and that C(39,3) equals 9139. Taking 10 cards from the remaining fifty cards means that we take away aces, kings, and two queens, leaving ten ranks lower than a queen.

Aha! For the special case when we, or anyone else, have pocket queens, 9720 flops have an ace, king, or queen, and 9880 flops have no rank higher than a jack.

What does all this mean to us players? It means we should expect just as many flops with aces, kings, and queens, as with queens, jacks, and lower ranks.

It means that when we raise forcefully with our pocket jacks and lower pocket pairs, we should expect to see overcards on the flop. Clearly, we should have a backup plan: (a) bull our way through anyway by betting and raising at every opportunity; (b) check the hand down; or, (c) fold at the first sign of opposing strength. Of course, our plan of attack also depends on the temper of our opponents.

As John Vorhaus wrote in Killer Poker No Limit!, page 161, "There are three ways to play pocket jacks - all wrong." Since half the flops have one or more cards higher than a jack, he's half right.

Mr. Burke is the author of Flop: The Art of Winning at Low-Limit Hold 'Em, on sale at amazon.com & kokopellipress.com. E-mail your Hold 'Em questions to richardburke@comcast.net