ddubois
09-22-2004, 07:31 PM
Question #1) I hold AB and try to steal a blind and am called. The board comes up CDE. What are the odds my opponent holds an C, a D, a E, or a pocket pair?
Question #2) I hold AB and try to steal a blind and am called. The board comes up CCD. What are the odds my opponent holds an C, a D, or a pocket pair?
(Basically, I want to know how likely it is that if I stab at the flop, my opponent will be forced to fold. I think knowing this information will give me more courage in post-flop play late in SNGs.)
I will make a feeble attempt at answering my own questions. I know that 5.8% of the time my opponent will hold a pocket pair. (Does the odds of this likelihood change given that I have seen ABCDE? I think it would, but the differnential is probably so small as to be insignifigant.) The 94.2% of the time my oppponent holds two unpaired cards, before the flop, we know it will miss him 1 - (44/50)(43/49)(42/48) = 32.4% of the time. But again, I think this changes given that we have seen ABCDE, because now we know that I didn't pair, and we know the board didn't pair. Both events make it more likely my opponent paired, so the 1 - (44/50)(43/49)(42/48) method is attacking the problem from the wrong direction. Ok, so let's start with, P(he has C). I think with 5 known cards, this is 1 - P(no C) = 1 - (44/47 * 43/46) = 12.5%. P(he has CD or E) = 1 - P(no C, D or E)? Is this 1 - (38/47 * 37/46) = 35%? Or is this wrong because they are not independent events? The resulting answer does seem reasonable however. If it is correct, then my hypothesis was right: the fact that ABCDE are exposed and known to have no pairedness amoung them does have a small impact on the likelihood of opponent having paired, but it's pretty inconsequential.
Now on to the CCD case. P(he has no C or D) = 1 - (42/47 * 41/46) = 20.1% so, unexpectedly, if the flop comes CCD, it's signifigantly less likely to have hit my opponent.
So the answer to the original question #1 is .058 * 1 + .942 * .35 = 38.8%. And question #2 is .058 * 1 + .942 * .201 = 24.7% (I think these are probably off by a little bit, again, the independence issue. But probably close anyway?)
Two followup lines of thought:
Question #3) How would I modify the problem to acocunt for the fact that people who call pre-flop raises on the bubble of a SNG are signifigantly more likely to have an ace than a ten, and more likely to have a ten than a 2? And these people are signifigantly more likely to hold a paired hand than an unpaired one? There is some bayesian math involved here, yes? I'd love to see some math or a formula that takes this into account. (Is there software I should own applicaable to this problem?)
Question #4) Any opinions on what conclusions I should I draw from this, with regards on how to play late in a tournament where the blinds are big?
(FYI, the specific scenario that got me the make this thread was that I raised UTG 4-handed a sixth of my stack on a blind steal with KQ, the flop came 339, and it was checked to me. I bet less than half of the pot, and he check-raised me all-in, either because he correctly sensed my bet as weakness, and/or because he had a pair/a big ace he thought was the best hand. I folded and greatly regretted not taking a free card to catch a K or a Q -- but had I missed and been bet off the pot on the turn, I am sure I would have regretted showing weakness by checking the flop. Stabbing at a pot like this has a big cost associated with it, I want to gauge how often it has to work to be profitable. Maybe if I bet larger he folds more often, or maybe he hit something and I was screwed regardless.)
Question #2) I hold AB and try to steal a blind and am called. The board comes up CCD. What are the odds my opponent holds an C, a D, or a pocket pair?
(Basically, I want to know how likely it is that if I stab at the flop, my opponent will be forced to fold. I think knowing this information will give me more courage in post-flop play late in SNGs.)
I will make a feeble attempt at answering my own questions. I know that 5.8% of the time my opponent will hold a pocket pair. (Does the odds of this likelihood change given that I have seen ABCDE? I think it would, but the differnential is probably so small as to be insignifigant.) The 94.2% of the time my oppponent holds two unpaired cards, before the flop, we know it will miss him 1 - (44/50)(43/49)(42/48) = 32.4% of the time. But again, I think this changes given that we have seen ABCDE, because now we know that I didn't pair, and we know the board didn't pair. Both events make it more likely my opponent paired, so the 1 - (44/50)(43/49)(42/48) method is attacking the problem from the wrong direction. Ok, so let's start with, P(he has C). I think with 5 known cards, this is 1 - P(no C) = 1 - (44/47 * 43/46) = 12.5%. P(he has CD or E) = 1 - P(no C, D or E)? Is this 1 - (38/47 * 37/46) = 35%? Or is this wrong because they are not independent events? The resulting answer does seem reasonable however. If it is correct, then my hypothesis was right: the fact that ABCDE are exposed and known to have no pairedness amoung them does have a small impact on the likelihood of opponent having paired, but it's pretty inconsequential.
Now on to the CCD case. P(he has no C or D) = 1 - (42/47 * 41/46) = 20.1% so, unexpectedly, if the flop comes CCD, it's signifigantly less likely to have hit my opponent.
So the answer to the original question #1 is .058 * 1 + .942 * .35 = 38.8%. And question #2 is .058 * 1 + .942 * .201 = 24.7% (I think these are probably off by a little bit, again, the independence issue. But probably close anyway?)
Two followup lines of thought:
Question #3) How would I modify the problem to acocunt for the fact that people who call pre-flop raises on the bubble of a SNG are signifigantly more likely to have an ace than a ten, and more likely to have a ten than a 2? And these people are signifigantly more likely to hold a paired hand than an unpaired one? There is some bayesian math involved here, yes? I'd love to see some math or a formula that takes this into account. (Is there software I should own applicaable to this problem?)
Question #4) Any opinions on what conclusions I should I draw from this, with regards on how to play late in a tournament where the blinds are big?
(FYI, the specific scenario that got me the make this thread was that I raised UTG 4-handed a sixth of my stack on a blind steal with KQ, the flop came 339, and it was checked to me. I bet less than half of the pot, and he check-raised me all-in, either because he correctly sensed my bet as weakness, and/or because he had a pair/a big ace he thought was the best hand. I folded and greatly regretted not taking a free card to catch a K or a Q -- but had I missed and been bet off the pot on the turn, I am sure I would have regretted showing weakness by checking the flop. Stabbing at a pot like this has a big cost associated with it, I want to gauge how often it has to work to be profitable. Maybe if I bet larger he folds more often, or maybe he hit something and I was screwed regardless.)