Fact-checking ESPN again

OK, I guess you all know the drill by now: ESPN broadcasts the World Series of Poker and includes a "poker fact" or two each week. I check them for accuracy, and find that the network production team is unforgivably sloppy.

Here's the first one from Tuesday night:





Not surprisingly, this appears to be wrong.

Here's how I figured it. It's actually easier to calculate the probability of not hitting a pair, then subtracting that number from 1. So all three flop cards have to be something other than a pair to my two hole cards. Let's say I have A-K to start with, and consider the first card on the flop. There are 50 unknown cards left in the deck, of which 6 are the remaining aces and kings, leaving 44 others. Therefore, the probability of the first card not pairing me is 44/50. If that happens, then by similar reasoning the probability of the second flop card not pairing me is 43/49. If that happens, then the probability of the third flop card not pairing me is 42/48. Multiply these together (because they all have to be simultaneously true), and we get 0.6757, or 67.57% as the probability of not catching at least one pair on the flop. Therefore, the probability of catching at least one pair (and it could be two pairs or trips or a full house or quads--doesn't matter for this calculation) is 1.00 - 0.6757 = 0.3243, or 32.43%.

This is confirmed in Phil Gordon's Little Green Book, p. 272 (though he again adds that superfluous and incorrect extra digit, and reports the figure as 32.40%) and in Matthew Hilger's Texas Hold'em Odds and Probabilities, p. 186. This gives me considerable confidence that the number is correct.

This time, it appears that the ESPN crew simply introduced a typographical error. I can't imagine any alternative way of thinking about the question or phrasing it that would result in an answer of 34.43%, and the match of the two figures after the decimal point leads me to suspect a simple digit error. Somebody looked at "32.43%" on a piece of paper, and mistyped it in the television graphics editing software as "34.43%." That's certainly an easy error to make, but it also should have been an easy one to avoid, or at least catch before it hit the air.


Our second "fact" for the week is this:





This is mostly correct, but needs a bit of clarification.

You can find a handy list of starting hands, ranked by their probabilities of winning against nine random hands (i.e., a 10-person poker table) here. Looking down the list, you can see that ESPN's top five are correct, as long as you specify that the AK has to be suited. If it is unsuited, it drops down to 12th in frequency of wins.

ESPN should also specify a full table, because these rankings change as you get short-handed. In the extreme, when playing heads-up poker, the ranking morphs to this:

AA 0.85
KK 0.82
QQ 0.80
JJ 0.77
TT 0.75
99 0.72
88 0.69
AKs 0.67
77 0.66
AKo 0.65

You can see that AK now drops below a bunch of pocket pairs over which it would be ranked higher in a full ring game. (Figures taken from Hilger, pp. 226-237.)

All in all, I would score ESPN this week as one wrong, one mostly right but incomplete. I suppose that's about as good as I've come to expect from them.


Addendum

Every time I do one of these math posts, I get something wrong. Fortunately, I have my eagle-eyed readers to catch what I miss, so that I have a chance to set it right.

This time, somebody emailed me privately to note that the Wizard of Odds table to which I linked above is presented in order of amount of expected win, not probability of win, even though the latter is the first column shown. I overlooked this. The two orders are not quite the same. For probability of win (or tie) in a ten-handed game, Wizard of Odds gives these for the top ten hands, in order:

AA 31.36%
KK 26.43%
QQ 22.66%
AKs 21.73%
AQs 20.44%
JJ 19.84%
KQs 19.8%
AJs 19.51%
KJs 18.94%
ATs 18.87%

To check this, I ran a simulation on PokerStove, using the specified hand against nine random hands in a Monte Carlo run of one million hands each. What I got was somewhat different, both in numerical values and order, from what Wizard of Odds shows (I'm using the "equity" numbers here, though the "probability of win" comes out in the same order as "equity"):

AA 31.1%
KK 26.1%
QQ 22.2%
AKs 20.7%
JJ 19.3%
AQs 19.2%
KQs 18.6%
AJs 18.2%
KJs 17.7%
ATs 17.4%

In other words, PokerStove agrees with what ESPN displayed (as long as we agree that the AK has to be suited), but Wizard of Odds finds that JJ and AQs flip for the #5 and #6 spots. I really don't have any plausible explanation for the discrepancy, and will have to leave it as a little mystery for now. However, I have enough confidence in PokerStove that if I were forced to guess which is correct, that's where my money would go.