Chances Are: Part VII

You frequently need to know how the chances of two or more events combine. On the flop, you need a heart for a flush. You might get it on the turn, but if not you might get it on the river. A slight variation is the case when you have a backdoor heart flush (only three of your suit after the flop). You will make your hand only by hitting a heart on the turn and then hitting another heart on the river. You are thinking about bluffing on the river. Suppose you are against only one opponent, then you need to know the chance he’ll fold. What if there are two or even three opponents? Now you need them all to fold. In this column I will examine these situations and give you the procedure for making these types of calculations.

Let’s take these situations in order. There are two ways of looking at the case in which you need a heart on either fourth or fifth street. One way is to add the chance of a heart on the turn to the chance of a heart on the river. Each of these will happen about 20 percent of the time, so it would appear that you would make a flush 40 percent of the time. This will overstate your chances, however, since you will occasionally hit a heart on both streets, but can only win the pot once. Since this happens about four percent of the time, we must subtract it. Forty-percent minus four percent equals a 36 percent chance of making a flush. The other way to approach this type of situation is calculate your chance of missing twice. Since something happens every time, the chance of making a flush is equal to the chance of missing subtracted from 100 percent. In this case, you miss the first time 80 percent of the time. Of that 80 percent, you will miss again 80 percent. So two misses equal .8 times .8 or .64. If you miss 64 percent then obviously you hit 36 percent, which is the same figure we arrived at above.

Now let’s look at the case were you do need two hearts in a row. It is fairly tough to hit runner-runner. Again, rounding to make calculations easy, you will hit the first heart 20 percent of the time. Then you will hit the second heart 20 percent of that 20 percent. This means that you will make a backdoor flush around four percent of the time. We can summarize all this by looking at the turn and river combined. Sixty-four percent of the time there will be no hearts. Thirty-two percent of the time there will be exactly one heart. Finally, four percent of the time both cards will be hearts.

Notice that the chance of something happening or not happening twice in a row is the chance of it happening once squared. Two misses are .8 times .8, equaling .64. Two hits are .2 times .2, equaling .04. Which brings us to the third question. When there is only one opponent, you have to estimate the chance of his folding compared to the pot. For example, if you make a pot-sized bet, you risk losing a pot-sized bet and your gain, winning the pot, are equal. If he folds more than half the time you will show a profit. Let’s say he folds 70 percent of the time and that for easy calculation, the pot is 100. You win .7 times 100, and lose .3 times 100. Seventy minus 30 equals 40. A very profitable play. Now, let’s say that there are two opponents. You now need them both to fold. As stated in the beginning of the paragraph, this is .7 times .7. They will both fold only .49, or just less than half the time. This bet would show a tiny loss (.49 times 100 minus .51 times 100 equals minus 2). Obviously, it is impossible to know your opponent’s calling percentages this exactly. Making this bet, which is approximately a breakeven play, probably makes sense, since you will get a wild image if they call and win the pot if they don’t.

What about three opponents? With two we squared the chances (.7 times .7). With three opponents we have to cube the chances (.7 times .7 times .7). This is only equal to .34. This is a very bad bluff. It will clearly lose a lot of money (.34 times 100 minus .66 times 100 equals minus 32). Against one opponent you showed 40 profit. Against two you broke even. Against three you lost 32. This illustrates why it is very dangerous to bluff multiple opponents. Most players instinctively realize this and tend not to bluff in these spots. This means that you have to be much more careful about calling someone who bets into a lot of opponents. He’s probably not bluffing. ♠

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.