What are the odds?

What are the odds? It’s often the first thing I ask myself when I’m deciding what to do in a poker hand. Often, the odds dictate what to do. Because I play limit holdem mostly, decisions are usually simple. I either have the best of it if I call or I don’t. Play for a while and you have the most common odds memorised. Drawing to a flush, a bit more than four to one; drawing to an openended straight, 4.75 to one; drawing to an inside straight, about 10 to one; drawing to a set, eight to one on the flop, 22 to one on a single card. For the uninitiated, drawing simply means trying to get the card that will make your hand a more powerful one. If I have JT and the board shows 984, I have an openended straight draw. Any Q or 7 will make me a straight, which is usually a strong hand.

A couple of things confuse most players who are trying to play well. First, you haven’t actually won when you draw with the correct odds and miss. You have lost the bet you made. The reason you take bets when you have the best of it is that you will tend to win over the long term. But the long term can be very long! It is because of the law of large numbers that it is worth taking bets that favour you. The other thing is that the long term starts with your next hand; in other words, it is reset. Just as if you were to flip a coin and get ten heads, you will not be any more likely to hit a tail on the next flip. It is just that the ten flips will turn out to be a blip in the bigger picture. This is hard to remember when you have missed four flushes in a row! You expect to win the next. But you are no more likely to win the next than you were the last. Over time, your results will converge on the odds, ever closer with ever more hands, but the past is not a guarantee of the future in the short term. Which can be frustrating, of course. When you are running cold, not picking up hands with the distribution you would expect, it can be very painful to miss flushes over and over again. (What do I mean, the distribution you would expect? I mean that when you receive cards at random you can expect high cards x% of the time, pairs y% of the time, suited cards z% of the time, just because those cards make up those percentages of the combinations possible from 52 cards.) So why trust the odds? Why not gamble more often on my draws coming in? That’s what some of my opponents do. They try to draw to outrageous longshots, which sometimes come in. But they pay for that sometimes, over and over, with a bet here, a bet there. Mostly, they only count the ones they win, and do not think about all the dollars they threw away chasing when they had the worst of it. The reason I don’t do it is because I intend to play for a long time and I don’t want to go busto doing it. I can be confident that over time, the times my flushes come in will converge on their odds (they will, even in the shorter term, tend to come in as often as expected: anecdotally, I can report that I do seem to hit a flush about one time in five; sometimes, I’ll get two in a row, then another five later, then miss seven, but it seems to be about the right amount). There are a couple of further problems that players often ignore with draws, but I am aware of and consider carefully. The first is that when we calculate odds, we do it with all unseen cards. If at the turn, I hold two suited cards, and two more are on the board, I have 46 cards I haven’t seen. Nine make my flush, so I say I have odds of 37/9 to hit my flush (just more than four to one, as I noted above). However, it’s perfectly possible that there are not nine cards that make my flush, because one of the other guys might be holding one (indeed, he’s likely to). Of course, this is offset by the other guys’ also holding cards that are not flush cards. In the long run, it evens out (because the proportions of flush cards held by others to nonflushcards held by others will converge on the proportions that they “should” be holding). In the shorter run, if I feel someone else is drawing to the same flush as I am, my calculations are slightly complicated. I will tend not to chase the marginal draws in these circumstances. (You can take the idea too far though, because you cannot know what others hold.) The other issue is that you can make a draw but not get paid enough to have made it worthwhile (in poker terms, your implied odds were not sufficient).

Generally, when I chase the flush draw, it is the pot that is offering me the odds I need. If it contains six big bets and it is a big bet to me, I will be paid sufficiently (and I will be paid for the other four draws I missed in the same circumstances). But when, for instance, you play a small pair before the flop, you are hoping to make up eight bets or more (because it is eight to one that you will flop another of the type of card you have in your pair). Some players will raise smallish pairs to try to knock players out but this is a bad play because, on being called in a couple of places, they do not have enough opponents to pay them off if they hit and have too many to hope their smallish pair will hold up. (The corollary of this thinking preflop is that it is a mistake to call with low suited cards, which are about 16 to one to make a flush: you simply will not often enough win 16 bets to make up for the ones you lose, not to mention that low flushes are more often beaten by bigger ones than flushes with a big card. Luckily, many of my opponents do not understand this and play any two suited. I can’t remember how often one is dealt suited cards but it’s quite often, something like one in four or five hands.) No limit is a bit different. The odds of drawing are the same but other variables come into play. If you have a small pair and can see a flop relatively cheaply, you are delighted. Whereas in limit holdem your implied odds are restricted by not being able to bet more than a bet or a double bet at a time, in no limit the potential take with a set is a whole stack. Players forget though that for you to get that stack your opponent must hit some kind of hand. This doesn’t happen often enough that you’ll want to call too many bigger bets before the flop. An example will show what I mean. With blinds at 50/100, a player in EP bets 200. Another guy raises to 500. I call with AJ suited (a pretty good hand). Both call. The guy in EP has paid five times the big blind to see a flop. He happens to have 55. The flop comes with a 5 and two other cards. Neither help me. He checks and the other guy checks. I check. The turn is a blank. The first guy bets out. The other guy (foolishly, he has nothing) calls. I fold. On the river, the guy with the set of fives bet again, not too heavily, and the other guy folded. My memory fails me as to how the action went precisely but it was something like that. The point is that I didn’t pay off and I saw that the middle guy had QJ. So the guy with 55 did well, right? Well no. He called a big bet preflop and got lucky. He was paid something like, let’s say, 2000 chips. But he will only hit that bet one in nine times. He paid 500 to drag 2000. That’s only four to one! He ended up taking the worst of it. That’s bad poker. The guy in question is one of the better players among those I play on Friday night but that kind of play makes it clear to me that he is not all that good: he has run hot a couple of nights rather than actually played particularly well (because any sucker can play a pair of fives like that and get lucky). If you bet 500 chips every time you have a small pair, you’re going to have to get paid 4500 when you hit. Because he doesn’t understand that (he thought he had the pot odds for a call: 300 more with 900 in the pot), I know he is not too sharp.