Conditional Probability

I recently played in a $20,000 buy-in no-limit hold'em invitational tournament at Wynn Las Vegas. It was a short field of 23 players and would air live on FoxSportsNet.



The structure was rather fast in order to get down to the final six in time for the live show. I ended up making the final table as the chip leader, but it wasn't a monster lead at all. We were all packed pretty closely together.



In the first 30 minutes of play as the chip leader, I didn't get even one hand I could remotely consider playing. Yet, I hung on to the lead, as the play was slow, for the most part.



John D'Agostino was on the short stack with $13,100 when this hand came down. The blinds were $600-$1,200 and "JDags" went all in for his $13,100. Next to the button, Ted Forrest contemplated for a moment or two, and then just decided to call the bet.



When it got around to me in the small blind, I found A-Q offsuit and was faced with my first dilemma: reraise to shut Ted out of the pot, or just call and hope to check the hand down with Ted. Knowing how tricky Ted can be, I couldn't rule out the possibility that he was setting a trap in this situation, so I decided to proceed cautiously by just calling.



The flop came 6-6-4 rainbow and I checked quickly, letting Ted know that I had no interest in bluffing into an empty side pot. Ted checked behind me and the turn card was an ace. With about $40,000 in the pot, I decided to throw out a dinky little goofy bet of $3,000. Why? Well, it's almost like a check, but it gives Ted a chance to fold a hand like 7-7 if he chooses to do so.



Ted called the turn and the river brought another 6, for a final board 6-6-4-A-6. I checked and made it rather obvious (at least to me) that I didn't like that card. I figured that my A-Q was probably in the lead, unless of course Ted had A-K.



Ted started contemplating, and it reminded me of a situation he played against Antonio Esfandiari in the heads-up tournament on NBC. In that hand, both Ted and Antonio had a 6-high straight with a board of 2-3-4-5-K. Ted bet, Antonio raised, and Ted reraised Antonio all in, getting Antonio to fold his straight.



As Ted started reaching for chips, I said aloud, "You're not seriously thinking about going all in here, are you? I saw you bluff Antonio off the same hand on TV."



The comment seemed to puzzle Ted, and he decided to check. I turned up my hand, fully expecting to rake in at least half the pot, when Ted turned over … pocket aces!



What the … ? How could Ted check aces full in that spot? Had he bet all of his chips, I would have been forced to call.



Howard Lederer was doing the commentary for the show and I spoke to him afterward. Howard defended Ted's play by rationalizing, "Ted made a good check there. There was a fifty-fifty chance that he was walking into four sixes. Also, if you didn't have the ace or the 6, you couldn't call anyway. If you have the 6 and he bets, he might eliminate himself from the tournament."



What Howard neglected to factor in with his commentary was the conditional probability of the situation. When doing "poker math," it's important to not only look at the static probability of your opponent having a certain hand, but to also factor in the likelihood that your opponent could hold the hand you are worried about.



In this case, I called a large all-in bet from the small blind. What hand could I call with there that contained a 6? You could make a case for me possibly calling that large raise with 6-6, or maybe even A-6. However, since there are already three sixes on board, I couldn't possibly have started with a pair of sixes.



I feasibly could have called such a large raise with A-6, but for you to believe I'd make a call like that, you'd have to assume that I didn't know it was a bad call.



Realistically, there was a 0 percent chance that I had a 6 in my hand. So, while Howard described the possibility of me having an ace or a 6 in my hand as fifty-fifty, since there was one 6 and one ace left, it was in fact impossible for me to have that 6.



Sure, it's true that I'll often play suited connectors in big pots because of the implied odds they offer, but in this situation, there was an all-in player and a dry side pot. Even if I hit a 6-5 of clubs, for example, I couldn't get paid off, since Ted would know that it would be silly for me to bluff in this situation.



Ted should have bet that river card, and he knows it. It was much more likely that I had the case ace in my hand than the 6. If I had that ace, he had me dead in the water and could severely cripple me with a bet; not to mention the fact that if I did somehow miraculously have that fourth 6, why would I check the river? Wouldn't I most likely bet and hope to get called by an ace?



I often make wisecracks about "math guys" and their approach to poker. Now, I fully understand how important math is at the poker table, but if you focus only on the numbers without factoring in your opponent's mood and playing style, you'll come up with the wrong answer far too often.



Poker is a game of people, first and foremost. You could have all of the "poker math" down pat, but without the ability to read into how your opponents think and what they are likely to do in most situations, all of that knowledge becomes worthless to some degree.



Players like Phil Ivey, Erick Lindgren, John Juanda, and Jennifer Harman all understand the mathematics behind the game, but they rely heavily on their ability to read people in crucial situations. Their thought process generally isn't, "Well, there is $600 in the pot and he's bet $200. That means I'm getting 4-1 odds that he's bluffing. I have no choice, I have to call."



It's more like, "Does this guy have it or not? Does this guy have the guts to bluff me here? Would he even play that hand in this spot? If he did have me, would he bet $200 or would he bet more?"



Of course, they would definitely factor in the fact that they are being laid 4-1 odds on the call, but not before first doing some exploration and looking for clues that will help them make a more informed educated guess.



Conditional probability is what you should focus on if you want to reach the highest levels in poker. Static probability can get you only so far.