Practical Probability - Part III

My last column looked a bit at the topics of outs and odds. In brief, outs are the cards that will win for you. If there is more than one card to come, outs are sequences that will win. In some cases, some of your outs will lead to a tie, in which case they can be counted as half of an out. For example, you have the A J and your opponent has the J 10. On fourth street, the board is 9 8 7 2. You have nine pure outs or winners (any spade), and two outs to tie (the 10 and 10). What about the other tens? The 10 has already been counted as one of your spades and he has the 10. Thus, you have nine full outs and two half-outs, or a total of 10 outs.



Remember that you must then use the number of outs to determine your chance of winning, and compare that to the amount you will win or lose. In the simplest case, one of you will be all in after his bet and your call. In that case, you need to weigh the odds you are getting from the pot against your odds of winning the hand. A hand with 10 outs will win approximately 20 percent of the time. It is a 4-1 underdog. If you are getting more than 4-1 pot odds, you should call. This will almost always be the case in limit poker. It is much less likely to be true in no-limit or pot-limit games. When your opponent bets the pot, you are getting 2-1. When he bets half of the pot, you are getting 3-1. In both cases, the odds are not sufficient enough to justify a call with 10 outs.



Converting Outs to Winning Percentages: There is a simple approximation for converting outs to a winning percentage – "The Rule of 2 and 4." With one card to come, multiply by 2; so, 10 outs will win around 20 percent of the time. With two cards to come, multiply by 4; thus, 10 outs will win approximately 40 percent of the time. How accurate are these approximations? Let's go through the actual calculations and see. The following material may seem boring or trivial, but it is worth your time and effort to follow along, since it will show you how to do this type of math and give you a better gut feel for the way these things work. Let's begin with the example above. Eight cards are known (your hand, your opponent's hand, and the board), so 44 cards remain in the deck. You will win 10/44ths of the hands, or around 23 percent. If you didn't really know your opponent's cards, you would win 10/46ths of the hands, or around 22 percent. Note that these figures are fairly close to the 20 percent suggested by The Rule of 2 and 4.



Now let's look at the case in which there are two cards to come. If you don't know your opponent's hand, there are 47 unknown cards. You will hit your hand on fourth street 10/47ths of the time, or about 21 percent. But of the 79 percent of the time that you miss, you will hit your hand 10/46ths of the time. So, your total winning percentage will be around 38 percent (this is fourth street's 21 percent plus the river's 79 percent multiplied by 10/46, which is about 17 percent). Again, this figure of 38 percent is quite close to the estimate of 40 percent that you'd get by using The Rule of 2 and 4.



One Card to Come and More Money: In the earlier example, we looked at a case in which, with one card to come, someone was all in on fourth street. In these cases, it is easy to compare the pot odds to your winning chances. It gets a little more complicated when there will be more betting on the river. In this case, the amount that you can win on the river must be added to the money that's already in the pot. In our example of a bet of half of the pot, you were getting 3-1 odds. For example, if there was 1,000 in the pot and your opponent bet 500, you would be getting 1,500-to-500, or 3-1. When there is more money to be bet on the river, you have the potential to win more. Presumably, you can't lose more, since you will fold if you miss your 10 outs. Let's say you think that you will win an average of 1,000 more when you make your hand. Now you are risking 500 to win 2,500. You are getting 5-1. This is more than the 4-1 that was the minimum you needed to call.



Two Cards to Come and Someone is All In: When making decisions on the flop, there are two cards to come. If there are two cards to come and someone is all in, the comparison is relatively simple. If you have 10 outs and your opponent goes all in with a half-pot bet, you are getting 3-1. With two cards to come, you will win around 40 percent of the time, and therefore will lose around 60 percent of the time; 60 percent divided by 40 percent is 3-2, and since you are getting 3-1, it is an easy call.



Two Cards to Come and More Money: Things get a lot more complicated when there is more money to be bet after you call the bet on the flop. Now there is a lot more that you can win if you hit one of your outs and your opponent calls your bets on fourth street and/or the river. Since it was an easy call when you couldn't win any more money after you hit, it might seem to be clear to call when you can. Unfortunately, there is a catch. You will hit an out on fourth street only 20 percent of the time. The other 80 percent of the time, your opponent may bet. In fact, he can bet enough to force you to fold. When he does this, you lose your chance to hit on the river. Let's return to our example hand and see how this happens. You have the A J and your opponent has the J 10. The flop is 9 8 7. There is 1,000 in the pot and you both have 1,500 left. He bets 500 and you call, thinking you will win around 40 percent of the time. Unfortunately, you miss on fourth street, so now the board is 9 8 7 2. The pot is 2,000, and he now bets his last 1,000. For a call to be correct, you would need to win at least 25 percent of the time. Since you will win only 20 percent of the time, it is clear to fold. What happened? His bet forced you to abandon your chance of hitting on the river. On the flop, you should have anticipated this possibility and folded instead of calling.



You had another choice on the flop, as well. You could have moved in yourself. Would this have been a good idea? Let's examine the odds. You would have risked 1,500 to win 2,500. You will win 40 percent of the time, so you are getting the right price to make this play. We can even see how much you make on this play; 60 percent of the time you lose 1,500, and 40 percent of the time you win 2,500. Your equity is 0.4 times 2,500 minus 0.6 times 1,500, or 1,000 minus 900. Your equity is 100 by moving in on the flop. In real life, this play would be even better. You don't know your opponent's hand. He won't always have the nuts, and you can expect him to fold some of the time. (This type of play is known as a semibluff. You are hoping he folds, but you still have winning chances if he calls.) In the next column, I will look more at equity and semibluffing.



Steve "Zee" Zolotow, aka The Bald Eagle, is a successful games player. He currently devotes most of his time to poker. He can be found at many major tournaments and playing on Full Tilt, as one of its pros. When escaping from poker, he hangs out in his bars on Avenue A – Nice Guy Eddie's on Houston and Doc Holliday's on 9th Street – in New York City.