Saturday, February 4, 2012

Poker Math: Pot Odds, Hand Odds and Implied Odds

Poker Math: Pot Odds, Hand Odds and Implied Odds
Let me first start off by admitting that math is not my specialty. In fact I think I only took it to the grade 11 level. It isn't that I am terrible at it, I just am not that well versed in it.

I just know the basics like most people. And as it relates to poker I do use some math but I guess I would classify myself more as the "feel" or "intuitive" type of player.

I have never thought that success in no limit hold'em cash games truly requires that much in depth knowledge of the math though. I think that as long as you understand a few of the basics such as pot odds, implied odds and hand odds, you will be fine.

And the easiest player for me to point to to prove my case is Dusty "Leatherass" Schmidt. He has famously made millions grinding online mid and high stakes cash games. He has said on numerous occasions that he doesn't really approach the game from a mathematical perspective either. He made these comments for instance a little over a year ago on his Cardplayer blog,

"I am a very intuitive player. I have only on and off used a HUD when playing and make a lot more plays by feel more so than by numbers."

Now I wouldn't go that far. I use my HUD all the time. Keep in mind though that he plays high stakes cash games where the player pool is much smaller than in the microstakes games that I play in. Therefore it is much easier to just go off memory. I don't really use my HUD with the regulars that I have plenty of history with at the micros either. So I can understand where he is coming from.

Basically what I am trying to say here though is that you don't need to get bogged down with stats and odds and percentages to be successful at this game. It works for some people and that is fine. Some people love to Pokerstove huge ranges and use all sorts of equity analysis tools. There is nothing wrong with that. But in my opinion you will be fine if you just stick to the basics as well.

I don't want to talk about HUD setup in this blog entry. Yes I am a minimalist but I absolutely think that you should use one and make frequent reference to it at the micros. For more information on the stats that I use and how to set up your HUD please refer to a pair of articles that I wrote on this subject last year (Part 1 and Part 2).

So I already mentioned above that I think that you should know some basics about pot odds, implied odds and hand odds. Where did I learn about this stuff? I ordered The Theory of Poker by David Sklansky from the Pokerstars FPP Store when I was first starting out. It was so long ago that I read this book that I hardly even remember much of it but from what I hear it is actually mostly about limit hold'em.

As if I knew any difference back then!

I think I might have actually been playing limit hold'em at the time though (yes I had a brief career in it lol). But also I have heard that it is actually a very difficult book for beginners.

So I am not specifically recommending that everyone goes out and buys it. I do think it is a legendary book in the poker world, and that is why it is one of the few that I recommend. However a lot of the information might be dated now and frankly a lot of the math in that book you could learn from a simple google search and a couple hours of reading. But it covers the basics well and organizes it nicely.

So let's talk about some of the ideas that it covers.

Pot Odds

What are they? Pot odds are basically the price that you are getting on the pot. Poker is often referred to as gambling of course and that is because there is both risk and reward in almost every action that you take. When the pot has a certain amount of money in it and you are calling a bet often it is with an inferior hand which you hope will be a superior hand should the right cards come your way on future streets.

So you look at the current size of the pot plus the bet in front of you (always remember that once somebody makes a bet in poker it is part of the pot) and determine if it is worth your while to call. That is, do you have the correct mathematical odds to call.

Example #1:

There is $1 in the middle and your opponent has bet $1. You have to call $1 in order to win a $2 pot ($1 in the middle plus his $1 bet). In order to get your pot odds you simply take the size of the pot and divide it by what you have to call. In this case 2 to 1.

Example #2:

There is $3 in the middle and your opponent bets $1.50.

$3 + $1.50 = $4.50,

$4.50/ $1.50 = 3 to 1 pot odds.

Hand Odds

So how do these pot odds relate to our decisions at the poker table? Well you take those pot odds (2 to 1, 3 to 1 etc) and compare them to your odds to make your hand. Now there is an important point to be aware of here. Should you make you hand you need to be highly certain that it will be the best hand. This is why for instance you should always try and make sure that your draw is to the nuts or near it.

There is a whole other type of odds in poker referred to as reverse implied odds which while outside the scope of this article basically refers to getting yourself into a situation where should you make your hand, it will be second best. Therefore your "prize" for making your hand is to lose a big pot. Fish are experts at this.

Back to hand odds however. So the easiest way to find out what your hand odds are is to find a chart on the internet. I just did a simple google search for "poker outs" and found this one,

What this chart will tell you is what your odds against hitting your hand are on the flop and turn depending on the number of outs that you have. Now an in depth discussion of counting outs will also be outside the scope of this article but it basically just refers to finding the number of cards remaining in the deck that will improve your hand to the best one.

You should spend some time and commit all of the outs and percentages with various draws to memory. So for instance a gutshot always has 4 outs, an open ended straight draw has 8 etc. And then make sure you roughly memorize the percentages to hit.

Make sure that you remember that there is a big difference between 2 cards to come (the flop) and 1 card to come (the turn). Many people make the mistake of forgetting just how much their equity decreases on the turn as compared to the flop. This is why it is often much better to play your draw in an aggressive manner on the flop than on the turn.

So how do we tie all of this together? Well we can make a simple comparison of our pot odds as compared to our hand odds and make some easy decisions.

Example #1:

You have a flush draw on the flop and your pot odds are 4 to 1. Since we know that a flush draw on the flop is roughly only a 3 to 1 dog to get there by the river we can profitably call in this spot because we are getting better than those odds on the pot.

Example #2:

You have bottom pair and a flush draw on the turn and your pot odds are 2 to 1. Since we know that our hand has roughly 30% equity or is a nearly 3 to 1 dog versus a top pair type hand we know that we cannot profitably continue in this spot. The price that we are getting on the pot just isn't good enough when compared to the odds of making our hand.

Implied Odds

Now if we were playing limit hold'em this article would basically be over at this point. But the great thing about no limit hold'em is that the stacks are often really deep and of course you can bet as much as you have in front of you at any given time.

So while it is not an exact science we need to roughly ask ourselves how much we think we can get out of our opponent should we hit our hand. Is this the type of opponent that is a strong player who is capable of making the tough fold when the obvious draw comes in? Or is it a recreational player who will perhaps moan and complain but pay you off in the end?

If it is the latter we can probably depend on getting a big bet out of him on the river which might change our decision in example #2 above from a fold to a call. However versus the disciplined regular who will not pay us off, we should still fold.

Implied odds are a difficult thing to quantify like pot odds and hand odds because there are so many different factors involved. For instance a big consideration is the likely strength of your opponent's hand.

If you are up against the type of opponent that only bets when they have an extremely strong hand, then maybe you can make a fairly loose call with a longshot draw because you know that they will be in a very difficult situation (and likely to make the crying call) should you hit.

On the flip side, a big mistake that I see a lot of players at the micros making is misjudging their implied odds against aggressive opponents. For instance I am a big advocate of raising a ton of hands from late position. I just made two videos for my blog about it in fact (Part 1 and Part 2).

I am raising upwards of 50% of my hands in some cases from the cutoff and button positions. Set-mining me out of the blinds would not make much sense because the vast majority of the time I will have absolutely nothing to pay you off with when you finally hit.

Lastly, there are considerations with implied odds to be made concerning the deceptiveness and strength of your draw.

If you have,


for instance on a board like,


probably even the greenest of villains isn't likely to pay you off much because the draw is so obvious when it comes in. Also your draw is not to the nuts. Somebody could have 58 and you will lose a big pot should you hit.

However if you have a hand like,


on a board like,


your draw is much stronger because it is far better hidden when it comes in. You always want to  consider what your opponent can have as well. In this case there are many more likely two pair hands that he could have or will make should the 9 or A come (people just play big cards more often than small cards).

Finally, our draw is to the nuts both ways. And we have a chance to cooler somebody that has a jack by making a higher straight should the 9 come.

I hope this discussion has been helpful for you all. As you can see I am no math expert. I consider myself around average in my knowledge of it. And at the poker table I use it sparingly. However it is important to know a few of the fundamental principles. And you should try your best to know them by memory.

If you are off by a little bit don't worry about it though. A flush draw will roughly hit 1 in 3 times from the flop and 1 in 5 times from the turn. See that chart that I linked above if you are a true nerd and want to know the exact values!

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Poker Math: Pot Odds, Hand Odds and Implied Odds


  1. Hi,

    Nice article, I have a question tho :).
    When you write the #1 example (the flushdraw) you write that we gave a 3:1 chance of getting there if we see the river, that I agree with, but how can we justify a call on the flopp if it's not all-in?
    There will most likely be another bet on the turn so wouldn't it be more correct to count it as 1 card to come?
    I might be thinking about this in the wrong way, just a bit confused :D.

    1. Thanks and you make a good point. I don't think that we can always assume that there will be another bet on the turn though. There might be and there might not be. If we knew 100% that there will be another bet then I think we should always view the odds as you say as if it is just one card to come. Issues like implied odds will also play a role though.

  2. "Set-mining me out of the blinds would not make much sense because the vast majority of the time I will have absolutely nothing to pay you off with when you finally hit."

    Sorry, but I don't understand this. Why does your wide open range from the blinds mean that you won't pay off sets? Are you saying that you'll bet so much from the button pre-flop that the majority of your stack will already be on the table when the set hits? Even if that is what you're saying, the player with the set still gets your money, right? I'm confused.

    1. He means he will probably not have a strong hand. Once the guy in the blind starts showing aggression he will just fold.