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1. Odds
Some gamblers might have heard this word over and over again without really knowing what it meant. Odds are a cousin of probability. So, what’s probability? Probability is the chance that a given event will take place. When the weatherman says there is a 25 percent chance of rain today, he is expressing a probability. He is saying the probability that it will rain today is 25 percent. What that means is that if today happened 100 times, 25 of those times it would rain, and 75 times it wouldn’t. This brings us back to odds. Odds compare the number of times an event will happen to the number of times it won’t. In our weatherman example, the odds against rain falling today would be 75 to 25 — that is, for every 75 times that it would rain, it wouldn’t rain 25 times. We write these odds, 75-25. It is equivalent to express these odds as 3-1, because we can see that for every time it rains, it doesn’t rain three times (75 divided by 25 equals 3).
Let’s look at the probabilities and the odds for some different events.
Coin flip, heads: probability 50 percent, odds 1-1
Airline flight delayed: probability 12.5 percent (data from Bureau of Transportation Statistics), Odds 7-1
Picking the A out of a deck: Probability 1/52 = 1.9 percent, Odds 51-1 (in this case, it’s easier to do the odds)
1a. Implied Odds and Reverse Implied Odds
Implied odds (and reverse implied odds) are based on the possibility of winning (or losing) more money later in the hand. They consider the situation after the next cards have been dealt and explain situations where things are better (or worse) than pot odds make them seem. Put another way, implied odds is the ratio between the amount you expect to win when you make your hand (more than what is in the pot) versus the amount it will cost to continue playing. In contrast, reverse implied odds is the ratio between the amount in the pot (what you win if your opponent does not make their hand) versus what it will cost you to play until the end of the hand. One of the major factors behind considering implied odds is how hidden your hand is (how uncertain your opponent is of your hand); another is the size of future bets. For the latter reason, implied odds become more important in no-limit and pot-limit games than in fixed-limit games.
As an example of implied odds, consider that at the turn there is $12 in the pot, it is $4 to call (pot odds 3-to-1), hitting your hand means you very likely will win, and additionally your opponent is likely play to the showdown. If you miss you will simply fold (costing $4). If you hit you can expect to make an extra bet of $4 from your opponent, winning $16 total so your implied pot odds are 4-to-1.
For reverse implied odds, consider that you have a strong hand but little chance of improving and your opponent has a chance of improving to a hand stronger than yours, or possibly already has a hand stronger than yours (they have been betting and you are not sure if they are bluffing) - essentially a situation where you are not certain that you have the best hand. Say it is the turn and there is $12 in the pot and it is $4 to call (pot odds 3-to-1). If your opponent has a weak hand or misses their card they may stop betting in which case you would only win $12 (it costs $4 to find out you are winning). Otherwise, you have committed to playing to the end of the hand in which case it would cost you $8 to find out you are losing (pot odds 3-to-2). There are many variations to this scenario. The essential idea is that reverse implied odds should be considered when you are not certain you have the best hand; it will cost more in future betting rounds to discover this.
1b. Pot Odds
When you bet (or call a bet) you are, of course, trying to win the money that is already in the pot. How often do you have to win to make this profitable? Clearly not every time - if it costs you 10 to bet (or call) and there is 100 in the pot, then you'd be able to be wrong 9 times out of 10 and still break even. This is the essence of the pot odds: You're paying a fraction to win a larger sum. If you're more likely to win than you have to pay, then your bet/call is a winning move in the long run.
Let's try one of the standard examples for pot odds: The flush draw.
First you need to consider your odds on hitting the winning hand. In the case of a flush draw on the turn in hold'em, you're getting about 4-1 (actually 37-9, since there are 37 cards that will "miss" you, and 9 that will give you the flush, but 4-1 is a good enough approximation) if the flush will be the best hand. This means that the pot odds need to be 4-1 or better in order to make your draw profitable. For instance, if your pot odds are 3-1 (paying 10 to win 30) you would get this Expected Value calculation:
(-$10 * 37/46) + ($30 * 9/46) = -$8.04 + $5.86 = -$2.17
What does this mean? It means that if there's only $30 in the pot and you have to pay $10 to win it, you'll lose on average a little over $2 every time you do it. Not a good thing.
What if the pot was $50?
(-$10 * 37/46) + ($50 * 9/46) = -$8.04 + $9.78 = $1.74
Here, you average $1 profit for every call you make.
Understanding the concept of pot odds is essential in order to play winning poker. Poker - especially limit poker - is taking a relatively small edge and repeating it relentlessly, over and over again, and making a profit from it. Making plays that don't pay off in the long run will instead turn that profit into a loss.
Having said that, let's look at that first calculation again. Is it really a $2.17 loss? Always? Well, that depends a whole lot on what happens after you actually hit the flush. And this moves us into the next concept: Implied Odds.
Where pot odds take into consideration the money that's in the pot right now, implied odds is an estimation on how much money you CAN win from the bet if you hit one of your outs. For instance, with 100 in the pot, and a bet of 20, is your gain really only 100 if you win? Can you really not squeeze out an extra few bucks from your opponent if you hit your flush? You probably can - and so as the pot will get bigger, your implied odds go up.
A good example of when implied odds come into play is when you limp in with a small or medium pair before the flop in hold 'em. Your chance of hitting a set (which is typically the only way a small or medium pair will win) is around 7.5-1, which means that pot needs to have 6 or 7 other limpers to make it worthwhile. But, of course, that's presuming that everyone will fold if you hit your set, which is rarely the case. Let's say instead that you get four other limpers and your bets will narrow the field down by 50% on the flop, and another 50% on the turn - what are your implied odds?
Four limpers to the flop = 4 SB.
Two callers to the turn = 2 SB.
One caller to the river = 1BB = 2 SB.
Here, you stand to win 8 small bets, at the initial price of 1, which gives your call positive expectation. By this count, your implied odds are good to make this pre-flop call with a weak pair because of the money you'll figure to win if you do hit your set, rather than the amount you're "guaranteed" to win.
Here's the downside to implied odds though: They're an estimation, and as it so happens, people tend to be way too optimistic in calculating them.
For instance:
K? 7?
On a board with:
Q? 9? 8? A?
gives you 9 outs to a flush, which is a 4-1 shot. Now let's say that there were only two of you in the pot, one limper and you in the BB. Flop was checked around, and he bet at the turn after you checked, the pot would be about 2BB. You're paying 2-1 to see the last card, which could give you the nut flush - but do you call? Pot odds say no. implied odds likely don't give you the numbers you're looking for either, but this is where people get overoptimistic!
If your opponent paired his ace and has no hearts, would he really bet into a four-suited board after the river? Would he call your bet? Probably not. You can hardly figure to win more than the money that's already in the pot at the turn, because if you make your hand on the river, he's not going to pay you enough. Even if he calls an extra bet on the river (maybe he has the J?), you're still not getting good enough odds. At that point, your call on the turn will have cost you 1BB, and you're looking at a profit of 3BB, which gives you 3-1. You'll have to successfully checkraise him (and he has to call your checkraise) for it to be near profitable, and you have to succeed at that every time that you hit your flush. Hardly likely.
I know some players who think this is mathematical mumbo-jumbo and has no place in a gambler's heart, but this is really the principle that separates winning players from losing players: Being able to tell a profitable bet from a non-profitable one. In the example above, there's a non-profitable bet being offered. Don't take it.
2. Combinations
In a game like Texas hold’em, we are interested in questions like, “what are the odds against completing a four-card flush draw after the flop?” This is a much harder question than, “What are the odds against completing a four-card flush draw after the turn?” In the latter case, there is only one card left to come. There are 46 unknown cards at that point (52 minus the two in our hand and the four on the board). So, to calculate our odds of making a flush draw after the turn, we just compare the number of unknown cards that don’t help us (37) to the number of unknown cards that do (nine). The odds of making a flush draw after the turn, therefore, are 37-9, or about 4.1-1.
After the flop, with two cards still to come, it’s not as straightforward. If we don’t make our flush on the turn, we could still make it on the river. How do we account for this? We do it by counting the different combinations of cards that could come. Say we hold the 9 8 and the flop is 10 4 2. The turn and river could be A K. They could be A A. They could be 3 3. They could be J J. Each of these is a different combination of turn and river cards. Note that J J is the same combination as J J, because they result in the same board. Now, instead of counting cards to determine our odds, we count combinations. If you write down every last possible combination for the turn and river in this hand, it turns out that there are 1,081. Then, if you look closely at all of them, it turns out that 378 result in a flush for our hand. So, the odds against making a four-card flush draw after the flop are 703-378 (because 1,081 minus 378 is 703), or about 1.86-1.
Just by learning these two terms, you now know how to calculate the odds against making any hold’em hand after the flop, or after the turn. Cool, huh? It is cool, but it’s also a lot of work to calculate your odds for every draw you might run into. Luckily, you don’t have to, as I’ll explain.
3. Outs
Your outs are the number of cards in the deck that will improve your hand. The flush draw we held above had nine outs. An open-end straight draw has eight outs. Two overcards have six outs. You could go through the odds calculation for each of these draws — or you could just read the results off the chart.
Again, it’s not important to know these exact numbers. In fact, there’s a useful trick called the Rule of Four to help you. Multiply your number of outs by four, and that number is roughly your percent chance of improving after the flop. So, with a flush draw on the flop, you have about a 9 times 4 = 36 percent chance of improving to a flush by the river (the actual number is 35 percent). Notice that this is the probability of improving, and not the odds against improving. Here are some quick conversions:
25 percent = 3-1 against
33 percent = 2-1 against
40 percent = 3-2 against
50 percent = 1-1 against
If you understand what I’ve written, you understand everything you need to assess your chances of improving in a hold’em hand. With enough practice, the numbers will become natural enough to you so that you can focus on other things at the table.
