A Math Trick: Negreanu & Ivey Bracelet Bet

Before this year’s WSOP, Daniel Negreanu and Phil Ivey were betting that they would win at least one bracelet. Was this a good bet? If they rate to win one or more bracelets at least 50 percent of the time, it is a good bet for them. (They may not need any more publicity, but this kind of thing does have a public relations value even if the actual bet itself is not plus expected value.) How could one go about calculating their chances?

There is a simple math trick that works well for these situations. Instead of calculating the chance that something will happen, calculate the chance that it won’t and then subtract from one. This may sound puzzling, so let’s begin with a simple example. You flop a four-flush, and want to know what chance you have of making it by the river. First let’s look at the hard way. You can make a flush on fourth street with nine cards. So you will make it on four, 9/47th of the time. 9/47 equals about .19. The remaining .81, you will make it 9/46th of the time. This is about .20, but remember it is only .20 of the times you missed. So we need to multiply .20 times .81, which is .16. Therefore you will make your flush .19 plus .16 or about .35. Now let’s use the trick. You miss on fourth 38/47 (.81) and you miss on fifth 37/46th of the time (.80.) Just multiply .81 times .80, and you miss a little more than .64. Therefore you hit .36 of the time. (The difference between the .35 and .36 comes from rounding errors.)

Now back to Daniel and Phil. They could conceivably play in 63 events each or a total of 126 events. To approximate their chance of winning a bracelet, we calculate the chance of their playing 126 events without winning. Assume they only win 1 percent of the events they play. This means they lose 99 percent. If they play every event, then their chance of not winning is .99 multiplied by itself 126 times or .99 to the 126th power. My calculator tells me that the chance of their not winning is .28. Therefore they will win a bracelet .72. This implies it is a great bet for them. Remember, however, we made some pretty liberal assumptions. They probably won’t play in 126 events. Even if they did, they will probably not win 1 percent of those events with large fields. These factors would make it much less likely that they will win a bracelet. But they will also play in some events with high buy-ins and short fields, such as the $50,000 Players Championship and $10,000 Heads Up. In these events their chance is much better than 1 percent. Overall I think they have slightly the best of it, but very slightly. The reason I think this is that their strategy will be devoted to winning. In tournament play, one frequently faces situations where there is one play that maximizes equity and another that maximizes winning chances. For example, if your goal is to make money, on the bubble it is generally right to be more conservative, and lock up a payday. If your goal is to win, you are happy to risk elimination short of the money in order to accumulate a big stack and have a good chance to win. ♠

Steve ‘Zee’ Zolotow, aka The Bald Eagle, is a successful gamesplayer. He has been a full-time gambler for over 35 years. With two 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.