Poker Math Lesson 101: Pot Odds and Counting Outs
What I will say is that most of the practical and relevant mathematics that apply to poker can be understood by almost anyone who is willing to invest a little bit of time and effort. Poker math is basically just applied arithmetic and is much easier to learn than most algebra, geometry, and calculus from your high school days. And there are many computer programs to assist you with the learning process, such as this site's free poker odds calculator.
Over the years, I've heard all the reasons why you don't need to learn math to win at poker. I've heard a lot of people defensively claim a lot of things to try to keep math from tarnishing the face of there precious and beloved game. These players convince themselves and try to convince others that poker is all about "the luck of the draw", or bluffing, or reading players. Well, I say let's learn poker mathematics and take their money.
When dealing with the mathematics of poker it is helpful to look at your poker career as a business. The easiest example to use is a casino. How does a casino run games of chance and make a profit? It's simple really; they offer patrons the opportunity to play games of chance that are statistically favorable for the house. Most players are happy to sit down at a gaming table and make bets that are mathematically erroneous mainly for entertainment. That entertainment has a cost attached to it. The cost is simply the statistical advantage or "edge" that the house has over them. Most poker players resemble casino patrons (in fact many are) in that they are playing mainly for entertainment. They are gambling. A player, who understands poker strategy and the mathematics behind it, can essentially take the place of the casino. Winning players make a profit from poker the same way the house does at the casino, by offering players opportunities to make and call bets that are statistically erroneous.
Poker is a game about decision making. Most of the mathematically defined decisions will be based on weighing your hand odds (your odds of having or making the best hand) against your pot odds (the amount of money in the pot compared to the amount of money it costs to contest it.). The decisions you make at the table will be correct or incorrect mathematically based on the correlation between your pot odds and hand odds. Each time you make a mistake you are giving your opponent(s) a statistical edge and each time your opponent(s) make a mistake you have gained a statistical edge. The edges that you take and the edges that you give away will determine your success in the game in the long term.
Calculating Estimated Value
By calculating the appropriate figures you should be able to get a pretty good idea of the EV (estimated value) of every play you make. EV is the amount of money a particular play earns or loses on average. To explain EV, I'll use a non-poker example; suppose you roll a standard die. If you were offered $10 every time you rolled a six and it only cost you $1 to play, you would be getting 10 to 1 on your money and the odds of you hitting a six on a standard die are 5 to 1 against. There are five numbers that will cost you money and one number that will make you money. One time in six, however, you will win $10 and 5 times in six you will lose $1. To find the EV of the play just subtract the money you lose from the money you win and divide by the number of trials. You win $10 and lose $5 ($10-$5=$5) and it takes you 6 trials ($5/6= $.86). That means our die rolling example has a positive EV of 86 cents. We will make, on average, 86 cents per trial. That's the way a casino looks at table games, that's the way you should look at poker.
Many of the difficult math problems that develop in poker occur when one player has a made hand and the other has a draw to a better hand. That's when a working knowledge of pot odds and counting outs will be most important, to profit when drawing and to make sure your opponents don't. I'll be using a hand from (NLH) No Limit Hold 'em as an example and referring back to it throughout the article. Let's suppose you start with AKc and you raise before the flop. Only the player in the big blind calls your raise. The flop comes out Qc 2d 8c, you've flopped a draw to the nut club flush. You have one opponent in the hand with you who you suspect has flopped a pair or better. If he bets out, should you raise, call, or fold? Most players will typically just check and call with a flush draw without ever considering whether or not it is actually a profitable decision. For simplicities sake we'll just assume that raising is not one of the choices, how do you determine whether to fold or call with a draw? The only way to decide whether to call is by comparing your hand odds to your pot odds. Supposing the pot contains $100 and your opponent bets $10, should you call with the flush draw? The short answer is yes, you're in "overlay". You're getting 11 to 1 pot odds, on a 4 to 1 draw. That's a very promising spread. You'll play.
Calculating Pot Odds
In the previous example, I said that the pot odds were 11 to 1 and the hand odds were 4 to 1. Let me explain those figures. The pot contained $100 before our opponent bet, his bet of $10 brings the pot size up to $110. It costs you $10 if you want to play. 110 to 10 is the same as 11 to 1. When you are getting such generous pot odds you will only need to be able to win the hand one time in twelve to break even. Anything beyond one chance in twelve is overlay, the positive EV you're looking for. So, if you can expect to win the hand better than one time in twelve you can afford to make the call. Now, I've estimated your odds of calling the bet then improving to the best hand at 4 to 1 against. Assuming you will improve to the best hand one time in five (4 to 1) when the pot is laying you $110 to $10 nets you an EV of +$12.00. That means you will win $12 on average every time you make the call. We arrive at that figure, again, by adding the money we lose together and subtracting it from the money we win then dividing by the number of attempts it takes us. We'll lose $10 on four attempts and win $110 on one attempt. ($110-$40= $70) We profit $70 dollars over five trials on average when we call this bet. $70/5 = $12 per trial.
If you count your outs (the number of unseen cards that will give you the best hand) and multiply them by two you will come up with your approximate chance in percentage of hitting one of your outs on the next card. It's actually a bit more accurate to multiply your outs by two and then add one. When chasing a flush draw you have 9 unseen clubs left in the deck that will give you the best hand. 9x2=18+1=19. You have about a 19% chance of hitting a club on the next card. 19% is about 4 to 1 odds. 20% is 4 to 1 exactly. If 20% of the deck will make you a flush, the other 80% won't. 80 to 20 is the same as 4 to 1. In our example, I said that I was estimating conservatively, and that's because I'm only counting the cards that are sure to give you the best hand. You also have the aces and kings in the deck that would give you top pair top kicker. There are actually three more kings and three more aces that may be outs. Depending on how tight or aggressive your opponent is, you might count those as ½ an out a piece. Your opponent could have a hand that is beating top pair, counterfeiting your kings and aces as outs. That's called discounting outs, I'll address it in more detail later in this series of articles on poker maths.
When drawing, it's important to remember to compare your chances of hitting your hand on the next card alone to your current pot odds when deciding if you should draw. That's because, if you call and miss your hand you will likely be facing another bet on the next card. I'm surprised how often a player will call a large bet on a draw when they have two cards left, then fold to the same bet if they miss the first card. The, somewhat fuzzy, logic behind this is in thinking that they have a good chance to improve with two cards left and then folding when they miss the first one. The truth is, of course, if you were going to fold your hand if you missed the turn then you didn't actually have two cards to begin with. More often than not the pot is actually offering better odds after the turn then after the flop. My point is, simply, that you must take poker one decision at a time. The only time you should weigh your odds of hitting an out with two cards coming against the pot odds is when one of you will be all-in after the flop. When that's the case you can go ahead and assess your chances of hitting your hand with two cards to come and compare them to the price the pot is laying you. And the easy, and approximate, formula is (again) your outs X2 + 1 per card.
Let's go thru another example; let's suppose you're in a $5-$10 limit hold 'em game. The blinds are $2-$5; you call in first position with 6c6d. Your call initiates a string of calls, then the dealer button raises. Everyone calls the raise so the pot contains $60 and is being contested by 6 players. The flop comes Ad 5h Tc, no help to you. With two over-cards to your pair you have to assume you're beaten. Everyone checks to the dealer who bets out $5. Action folds to you, what should you do? Well let's do some math. The pot is offering $65 to $5, which is a huge price, 13 to 1. You can figure to have two outs in the deck which only gives you about a 5% chance of hitting. You actually need about 19 to 1 to call here. 5% converts to 1 chance in 20. 1 chance in 20 converted over to odds becomes 19 to 1 against. Even if you count the implied odds (the bets you'll collect after you do make your hand), I'd say this is a fold. Some players might call the bet thinking that they can make a lot of money with the set if they do hit it. While the math might indicate that it is at least close to a call, you also have to consider that there are more players yet to act and any of them could put in a re-raise making it more expensive than you anticipated to continue to chase your set. All things considered, this is a fold.
In my next article on poker math we'll take a closer look at draws. We'll also examine the other side of the coin, protecting your hand against draws. We'll talk a bit more about implied odds and we'll try to establish some rules of thumb for the bet sizes you should be willing to call on certain draws, and the bets sizes you should select to protect your hand against draws. In closing I'd just like to remind you to be patient with all of this information. The more you study poker mathematics and apply it, the more useful it becomes. I used to practice counting my opponents outs even when I was not in the hand. The more you use the math the easier it becomes and the more money you'll make in the long run. I'm going to end with a list of draws, the number of outs, and the approximate percentages of hitting those draws per card. Good "luck"!
Poker Math Series by Dead Money
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