I'm watching this week's "World Poker Tour" (as the previous post probably suggested). Mike Sexton just said one of the dumbest things I've ever heard come out of his mouth.
Two amateurs are involved. One of them, Andrew, won a huge pot with K-K on the "previous" hand. (The reason for the scare quotes will become clear later.) Now a guy named Robert is on the button and raises with J-J. Andrew is in the big blind and reraises with 7-7.
While we're waiting for Robert to decide what to do, Mike Sexton says, "Well, Robert knows that he [Andrew] had two kings the last pot. You're never gonna put a guy on aces, kings, and queens back to back."
Does Mike Sexton actually believe that the fact that a player had K-K on one hand makes it mathematically less likely that he has A-A, K-K, or Q-Q on the next hand? If so, how in hell has he managed to make a living at poker for the last 20 years or so, with such a fundamental distortion in his grasp of randomness?
The cards have no memory. The auto-shuffler has no memory. What a player had on the previous hand has exactly zero effect on the analysis of what he might have in this hand. He is every bit as likely to have K-K here as if you wait 200 hands and then try to figure out what he's holding. When the K-K hand was over, everything reset, and he became just as eligible to get K-K again as he ever was.
Yes, it's rare to get K-K (or any other combination of cards you care to specify) in consecutive hands. For any given two-hand sequence you name, them both being K-K will happen only once in 48,841 times, on average. But the probability of the second hand being kings is completely independent of the probability of the first hand being kings. Once you know that the first hand was K-K, you know nothing more or less about the likelihood of the next hand being kings than you did before. It is 1 out of 221, just as it was on the first hand, and just as it will be on every subsequent hand. In fact, even if somebody miraculously gets K-K 10 times in a row, the probability of getting K-K on the very next hand, to make 11 in a row, is still precisely 1 in 221 (barring some shady dealing going on, obviously).
I've talked about this common fallacy at least twice before that I can remember, here and here. I'm not really terribly surprised when I hear a recreational player fail to distinguish between a priori probabilities (e.g., the probability that two specified consecutive future hands will both be K-K) and conditional probabilities (e.g., the probability that the next hand will be K-K, given that the previous one was), which is the essence of the mistake here. After all, our public school systems are about as lousy as they can be in teaching kids to understand statistics and probability.
But how can a guy who has spent so much of his life playing, thinking about, and talking about poker as Mike Sexton has still hold on to such a simple, rudimentary error in understanding how randomness affects the game? It's kind of like an obstetrician still believing that babies are brought by storks.
The other possibility is that Sexton understands this perfectly well, but is being condescending towards Robert--saying, in effect, "This guy isn't smart enough to understand that what Andrew had on the previous hand should not be taken into consideration when assessing his range of hands here." If so, then he's being unfairly demeaning to the amateur player, with no reason to make such an unflattering assumption about him.
There's another factor making this statement even more stupid, if that's possible. Because of editing, this was clearly not actually the hand immediately following the one in which the player named Andrew had K-K. I went back and checked where the button was, and it had moved two seats. So there was at least one intervening hand not shown, and maybe 7 or 13 or 19 hands (beause there are six players at the table) that were not shown. This doesn't affect the probability, of course, but it means that both the factual basis and the theoretical basis for Sexton's statement are wrong.
To his credit, I don't recall hearing Sexton say anything quite this boneheaded before. But it was sure a doozy.