To be a great poker player you must be able to combine mathematical knowledge, psychological insight, and the ability to read hands. This is especially true in No Limit Hold ’em especially when everyone has a lot of chips. Here’s an example:
You are first to act with one card to come against one opponent. There is $1,000 in the pot. You have a good hand, but you think there is a 20 percent chance his is better, where you are drawing dead. You are sure that your opponent will never bluff in the situation, and will always raise when he has it. Meanwhile the other 80 percent of the time you are quite sure he has a straight draw. 40 percent of his total hands are gut shots and 40 percent are open enders. You believe he will call a bet up to $400 with the gut shot, and up to $1,000 with the open ender. (He’s counting on implied odds, but he won’t get them since you won’t call this non-bluffer on the river.)
So how much should you bet into this guy? There is a 20 percent chance you should bet zero. A 40 percent chance you should bet $400, and a 40 percent chance you should bet $1,000. That averages out to $560. Could that be the answer? Unfortunately it’s not that simple. Say the gut shot is ten percent to bet, the open ender is 20 percent, and the other possible hand for your adversary is 100 percent for him. That means that if you check he will win the pot 32 percent of the time. (20 percent + 20 percent of 40 percent + 10 percent of 40 percent.) A check on your part (followed by a fold if he bets) thus gives you an expected value of 68 percent of $1,000 or $680.
Let’s see how various bet sizes do. How about $1,010? That forces him to fold his straight draw, but costs you all that money when he has his hand. Your expected value is 80 percent of $1,000 minus 20 percent of 2,010. That’s $598 which is worse than if you had checked.
What about $401? That folds his gut shot, which means that he wins 28 percent of the time. So your EV is 40 percent of $1,000 (when he folds) + 32 percent of $1,401 (when he calls and misses) minus 28 percent of $401. That’s about $736, the best result yet.
But what if we let the gut shot in and bet $400 exactly? Now he wins 32 percent of the time and it costs you $400 when he does. But you win $1,400 the rest of the time. 68 percent of $1,400 minus 32 percent of $400 is $824, which is still better.
Could a bet between zero and $400 be even more profitable? Hopefully you see immediately that it can’t be. Any such bet wins 68 percent and loses 32 percent, so the bigger the better and $400 is the biggest we can make it.
What about betting exactly $1,000? Like the $401 bet, it will fold the gut shot and get called by the open ender. That extra money is nice the 32 percent of the time the opponent calls and misses. But it costs you the times he makes it, plus the times he had it already. Let’s do the math. Your EV is 40 percent of $1,000 + 32 percent of $2,000 minus 28 percent of $1,000. $760 altogether, well below the EV of the $400 bet.
Could a bet above $1,010 be right? Of course not. Anything over that just wastes money, since it only gets called when you are beat.
Thus, the right bet is $400 exactly. Of course, in real life you could never be this precise. But this example illustrates an important general point. Namely that the right play is not necessarily the one that is most likely to be right. (Had I said a gut shot draw was slightly less likely than the open ender, the answer would not have changed.) $400 is the wrong bet if he has the hand and also if he has the open end straight draw. Yet it was clearly the right play overall. I leave it to the readers to mull over how the knowledge of this result might affect their No Limit Hold ’em game.
