Chances Are: Part III

Someone once joked that there are two kinds of people: those who are good at mathematics and those who aren’t. In most cases the math of figuring out the chances of an event happening and converting chances into odds is easy. The hard part can be figuring out the how many cases there are and how many include the event you are interested in. Here is a brief quiz.

1. With one card left to come the board has two spades. You have none, but your opponent has two. You know he can only win with a spade. He can only win if the last card is a spade. What is his chance of winning?
2. What are the odds against his winning?
3. You both have stacks of $500 and the pot is $500. If you move all-in what pot odds is he getting to call?
4. What should he do?
5. If instead of moving all-in, you bet $100, what pot odds is he getting?
6. What should he do?
7. How much should you bet?

The first six questions should be easy for you to answer.

1. Remember that the chance of something occurring is expressed as a fraction, a decimal or a percent. There are eight cards gone so there are 44 left. Nine of these are spades. He will win 9/44 or .20 or 20 percent (slightly rounded).
2. He will lose 35 times, so the odds against him winning are 35-to-9 or approximately 4-to-1.
3. If you bet $500, there will be $1,000 in the pot. It costs $500 for him to call. He is getting odds of 1000-to-500 or 2-to-1.
4. His hand breaks even around 4-to-1. Two-to-one is much less, therefore he should fold.
5. Now the pot is $600, but it only costs $100 to call. He is getting 600-to-100 or 6-to-1.
6. Six-to-one is much more than the 4-to-1 he needs to break even. He should call.
7. There is no easy answer to this question. I will try to explain why in the rest of this article.

Since he needs 4-to-1 odds to break even by calling, you should first figure out how much this is. Basically this means dividing the pot by three. $500 divided by three is $167. To check if this is correct: we figure that he would get pot odds of 667-to-167. This is approximately 4-to-1. Obviously, you want to bet enough so that he is making a mistake if he calls. You must bet at least $175. The more likely you feel he is to call, the more you should bet. I would think that a bet of around $200 has a good chance of being called. But notice that all of this is based on very specific assumptions. In reality neither of you really knows what exact cards his opponent has. Given that our assessment of the situation is pretty accurate, here are some slight variations.

Your Hand Board His Hand
A♥ Q♦ A♠ K♥ 8♠ 4♣ Q♠ 9♠
A♥ K♦ A♠ K♥ 8♠ 4♣ Q♠ 9♠
A♥ Q♦ A♠ K♥ 8♠ 4♣ Q♠ J♠
A♥ K♦ A♠ K♥ 8♠ 4♣ Q♠ J♠

I have kept the board the same in all four cases, but made slight variations to both hands. Let’s see how this chances things.

1. This one is the case we analyzed above. He has exactly nine winners.
2. This is a little better for you. Now the K♠ makes his flush and your full house. Now you can bet a little less. Not only does he only have eight winners, he has one disaster that will probably cost him a lot more.
3. Now it’s gotten better for him. Now he can win with nine spades and three tens. His chance of winning is much better. Also you now are in danger of losing more if a ten comes on the river. Now you must bet a lot more to insure that he is making a mistake by calling.

Now the K is again terrible for him. But a ten will make you lose and may cost you more money. You should definitely bet more than in case one, but not quite as much as in case three.

Notice how these small differences in the hands create huge differences in the correct play. That is why I said at the beginning that the mathematics were relatively simple. It is knowing the exact situation that is difficult. If you are feeling industrious, try to figure what are the odds for cases two, three and four. Then try to decide on an appropriate amount to bet. I’ll give my answers in the next column. ♠

Steve ‘Zee’ Zolotow, aka The Bald Eagle, is a successful gamesplayer. He has been a full-time gambler for over 35 years. With 2 WSOP bracelets and few million in tournament cashes, he is easing into retirement. He currently devotes most of his time to poker. He can be found at some major tournaments and playing in cash games in Vegas. When escaping from poker, he hangs out in his bars on Avenue A in New York City -The Library near Houston and Doc Holliday’s on 9th St. are his favorites.