Two weeks ago, when ESPN messed up on its weekly "Poker Fact," I thought it would be a one-time thing. They surprised me the next week by doing it again. I would not have believed they could hit the trifecta, but they somehow managed.
This week they say, "When holding any pocket pair, the probability of flopping a set is 11.76%."
Wrong.
The usual way of running this calculation, which is apparently how ESPN's people went about it, is to calculate the probability of the flop not having another of the rank in question. There are 50 cards left, so the probability of the first card of the flop not completing one's set is 48/50. If that happens, then the probability of the second card of the flop also not completing one's set is 47/49. If you have gotten that far without a set, then the probability of the third card of the flop still not completing one's set is 46/48. Multiple those three together (because all must be true simultaneously), and the probability of the flop not bringing a third card to one's pocket pair is 0.882449. Subtract that from 1.000000 (because the flop must either contain another of the rank in question or not), and you get 0.117551, or about 11.76%.
The problem, though, is that four of a kind is not a "set," and the calculation above does not exclude flopping quads. Given that you start with a pocket pair, the probability of flopping quads is 0.2449%, if I've done my math right.* (And if I haven't, please let me know in the comments so I can fix it.)
Therefore, the probability of flopping a set but not quads is 11.7551% - 0.2449%, or about 11.51%. What the ESPN graphic was showing was not the probability of flopping a set, but instead the probability of flopping either a set of four of a kind, starting with a pair in the hole. They are not the same thing.
Be sure to tune in next week for the next exciting installment of "How Many Ways Can ESPN Get the Math Wrong?"
By the way, when starting to write this post, I pulled out Phil Gordon's Little Green Book, which has a handy section of common poker probabilities. It's my usual first place to look for quick answers. But I see that he has it wrong here. On p. 271, he says that if one starts with a pocket pair, the probability of flopping a set is 10.80%, and the probability of flopping quads is 0.20%. Wrong on both points.
*We're specifying that the flop must contain two exact cards. There are therefore 48 cards left in the deck to fill that third spot on the flop. We don't care about the order, so that's effectively 48 different flops that will work. We already know that there are 19,600 possible flops, once we've accounted for the fact that our two hole cards can't be in the flop. 48/19,600 = 0.00244898, or 0.244898%.
Addendum
I just knew I'd mess something up. Every time I've tried doing a math post in the middle of the night (note the time stamp on it), I make a mistake. As one commenter pointed out, I failed to account for full houses. If flopping quads can't be counted toward flopping a "set," neither can flopping a full house. I even noticed Phil Gordon's book separating that out, but then my sleep-deprived brain dismissed it as irrelevant. (I thought so because what flashed through my mind was the pocket pair plus three of a kind on the flop, which I dismissed as not flopping a set.) Clearly it is not.
So what is the probability of flopping a full house if one starts with a pocket pair? Well, that means a flop containing a third of our pair's rank, plus another pair. Hidden among the 50 remaining cards are 12 full ranks. There are 6 different pairs that you can pull from a group of four: C(4,2) = 6. That means that there are 6 x 12 = 72 different pairs that can hit our flop. Each of those might occur with either of the two remaining cards that would make our three of a kind, for a total of 144 flops that meet our specifications for a full house. 144/19,600 = 0.007347, or 0.7347%.
We subtract that from the previously calculated 0.115102 and end up with 0.107755, or about 10.78% for the final probability of flopping a set.
So, to summarize--given a pocket pair, the probability of flops is as follows:
Set: 10.78%. (That's 1 in 9.28 times, or 8.28:1 odds against.)
Full house: 0.73%. (That's 1 in 137 times, or 136:1 odds against.)
Quads: 0.24%. (That's 1 in 417 times, or 416:1 odds against.)
Set or full house or quads (which is what ESPN's graphic was actually about): 11.76%. (That's 1 in 8.50 times, or 7.50:1 odds against.)
(Note that "full house" here refers only to pair in the hand, third of that rank on the flop, plus a pair of a different rank on the flop. You can also flop a full house with a pocket pair and flop of three of the same rank. But since this possibility does not include the third card to the rank of one's pair, it does not enter into the calculation here. For completeness, though: There are 12 full ranks left in the deck. Each rank has three different combinations that can make up a single-ranked flop, because C(4,3) = 4. So there are 12 x 4 = 48 possible flops that would be three of a kind on the board. 48/19,600 = 0.00244898, or 0.24%. By strange coincidence, this is the same probability as flopping quads, starting with a pocket pair.)
Phil Gordon gives these numbers as 10.80%, 0.70%, 0.20%, and 11.80%, respectively. This is a little peculiar, in that he appears to have done the math correctly, then rounded off to the nearest 0.1%, then later decided to make it look more precise and added in an extra 0 digit at the end. That last step is the part that isn't right. If he had left the numbers as 10.8%, 0.7%, 0.2%, and 11.8%, he would have been fine. My apologies for incorrectly faulting his set calculation in the original post. It appears that his only error was in rounding, then "unrounding" by adding zeros.
Wednesday, September 24, 2008
Three in a row???
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5 comments:
Should you also exclude full houses. Example: Your hole cards are 55. Flop comes 577.
Shane
ESPN hates poker, it's clear. They wish they didn't have to cover it, so they cover it by showing as little ACTUAL POKER as they can get away with.
The funny thing is, NBC seems to adore poker, shows as many hands as possible, and the babe who interviews the players at the end either ACTUALLY KNOWS SOMETHING about the game or is just fed REALLY sharp questions!
Furthering the hilarity, the way ESPN feels about poker, NBC apparently feels about the Olympics!
In my one day as King Of The World, along with other more pressing matters, I would absolutely find ten minutes to ordain that henceforth, ESPN may not cover poker and must cover the Olympics, and all poker coverage must be done by NBC.
Grump:
Actually, Phil Gordon and you are both correct. The probability of flopping a set or better is 11.76%, as you said. Then, you correctly wrote that that figure includes quads, which it does. That's another 0.25% (rounded to the nearest 1/100th percent). You were correct there.
But, you've missed something. As you and I both wrote, that 11.76% figure is the probability of flopping a set or better. "Better" also includes flopping a full house, which happens about 0.74%. So, that, too, must be subtracted.
So, you end up with 10.77% for flopping a set. That jives with Gordon's numbers (his is rounded up to 10.8). His 0.20% number for quads is wrong, though... so he must have either rounded down or gotten lazy.
Wouldn't it be a little higher in reality? What's the point of playing one handed poker? At a minimum shouldn't you take into account 2 other cards that will be out of play? Maybe assuming that the other player does not have one or both of the other two cards needed for the set?
Just a thought.
Chuck:
When I first started messing around with poker probabilities, I wondered the same kind of thing. But as it turns out, it doesn't matter how many other players there are. You can be playing all by yourself, pull a pair out of the deck, have literally all 50 remaining cards left, and the probability of the kind of flops described will be as I gave them.
Or you can have nine opponents, and the numbers will be the same. In fact, you could theoretically have you and 22 opponents (at a HUGE table!), so that 46 cards are gone, and only 6 left in the deck, and the flop probabilities would still be identical! It takes a while to wrap your mind around this concept, that's for sure.
It's weird but true. The basic reason is that when we don't know what those opponents have, it doesn't matter whether the cards are in their hands or in the deck.
Look at it this way: There really is no "probability" involved once the deck has been shuffled and cut. We either will or will not get a flop as specified; the die is cast. So we're not calculating the actual probability that the remaining cards will magically rearrange themselves at the last second to be in a particular configuration on THIS hand. Instead, we're saying that if you have a billion deals in which you start with a pocket pair, X percentage of those times you will get a flop of the given variety.
When you think of it that way, it's easier to understand why on any given hand it doesn't matter whether the cards we want to see are still in the deck or in an opponent's hand. Our probability numbers already account for the fact that they could be anywhere.
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