Learning The Required Poker Math

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There is a LOT of math in almost all games, and poker is no exception. That said, many players get bogged down in the math to the point that they completely ignore it or overly obsess over mathematical aspects of the game that do not matter at all (such as how often you will make quads and lose.)

Let’s explore various types of basic math you must fully understand if you want to have any chance to succeed at poker.

Pot Odds

Let’s first discuss the concept of pot odds, which allow you to determine how often you need to win a pot in order to break even based on the bet you are facing in relation to the size of the pot. The equation: % Required to Win = bet/(pot + bet + bet) can be used to determine how often you need to win in order to continue any time you are facing a bet when against one opponent.

Suppose your opponent bets $100 into a $100 pot. In this situation, you have to call $100 to win $300 total (the $100 pot, your opponent’s $100 bet, and the $100 you are putting in). This means you need to win the pot 33% of the time or more in order to profit 100/(100 + 100 + 100).

If you know it will win more than 33% of the time based on your hand’s strength or the likelihood that your hand improves to the best hand (which will be discussed soon), then you should continue in the pot. If it will win less than 33% of the time, you should consider folding.

Here is a chart listing various bet sizes you will face and the percentage of time you must win in order to profitably continue:

Bet You are Facing % You Must Win
10% pot .1/(1 + .1 + .1) = 8.3%
25% pot .25/(1 + .25 + .25) = 16.7%
33% pot .33/(1 + .33 + .33) = 20%
50% pot .5/(1 + .5 + .5) = 25%
67% pot .67/(1 + .67 + .67) = 28.7%
75% pot .75/(1 + .75 + .75) = 30%
100% pot 1/(1 + 1 + 1) = 33%
150% pot 1.5/(1 + 1.5 + 1.5) = 37.5%
200% pot 2/(1 + 2 + 2) = 40%
300% pot 3/(1 + 3 + 3) = 42.9%

You may be surprised to see that no matter how much your opponent bets, the most you will ever need to win is only 50% of the time in order to break even. That said, you will find that almost no one in the real world bluffs too often using a bet of three times the size of the pot.

When facing a bet with players yet to act behind you, you also have to consider how often you will be raised and forced to fold your hand. With players yet to act, you usually need to win far more than your pot odds dictate in order to continue.

Required Bluff Success Percentage and Minimum Defense Frequency

When you make a bet with a total bluff that cannot win when called, you can use the equation: Required Bluff Success Percentage = bet/(bet + pot) to determine how often your bluff needs to succeed in order to profit. Of course, you will rarely win 0% of the time when called when bluffing on the flop or turn, so you will usually need your opponent to fold less often than the required bluff success percentage in order to immediately profit.

For example, suppose you bet $300 into a $900 pot with a total bluff that will never improve. If your opponent will fold more than 300/(300 + 900) = 25% of the time, you immediately profit and therefore should bet with all of your bluffs.

When you are facing a bet, you need to defend at least some portion of the time; otherwise, your opponent will be able to profitably bluff with any two cards. This amount is the Minimum Defense Frequency, which is 1 – Required Bluff Success Percentage. Note that this is different from the percentage of time you need to win in order to profitably call. You need to consider both how often your hand will win as well as how often you should continue with your entire range.

While you should not always defend at or more than the Minimum Defense Frequency because you will under-realize your equity from out of position and your opponent’s range may be far stronger than yours (which I discuss thoroughly at PokerCoaching.com), it is an important concept to consider.

Here is a chart listing your bet size, your required bluff success percentage, and your opponent’s minimum defense frequency:

Your Bet Size Required Bluff Success % MDF
If you bet 10% pot .1/1.1 = 9% 1 – .09 = 91%
If you bet 25% pot .25/1.25 = 20% 1 – .20 = 80%
If you bet 33% pot .33/1.33 = 25% 1 – .25 = 75%
If you bet 50% pot .50/1.50 = 33% 1 – .33 = 67%
If you bet 67% pot .67/1.67 = 40% 1 – 40% = 60%
If you bet 75% pot .75/1.75 = 43% 1 – .43% = 57%
If you bet 100% pot 1/2 = 50% 1 – 50% = 50%
If you bet 133% pot 1.33/2.33 = 57% 1 – 57% = 43%
If you bet 150% pot 1.5/2.5 = 60% 1 – 60% = 40%
If you bet 200% pot 2/3 = 67% 1 – .67% = 33%
If you bet 300% pot 3/4 = 75% 1 – .75% = 25%

Determining The Number Of Outs

When you have a draw (a hand that is almost certainly behind at the moment but could improve to the best hand if the next card is favorable), you must figure out how many outs you have to improve.

If you have a gutshot straight draw (such as 6-5 on a 9-8-2 flop), you have four outs (four 7s).
If you have two overcards (such as K-Q on 8-7-2), you have six outs (three Kings and three Queens).
If you have an open-ended straight draw (such as 7-6 on 9-8-2), you have eight outs (four 10s and four 5s).
If you have a flush draw (two cards of the same suit in your hand and two cards of that same suit on the flop), you have nine outs (the remaining nine cards of the same suit).
If you have a flush draw and a gutshot straight draw, you have 12 outs (notice one of the straight cards also makes a flush).
If you have a straight flush draw, you have 15 outs (notice two of the straight cards also makes a flush).

It is important to note that all of your outs may not actually improve you to the best hand. When you have two overcards, such as J-10 on 7-6-2, if the turn is a 10, you could be drawing dead if your opponent has 9-8, 7-7, or 6-6 or way behind if they have 10-7, 7-6, or Q-Q. If you have a low flush draw such as 4 3 on Q J 6, if the turn is a spade giving you a flush, you could lose to numerous better flushes such as A 2 and K 8. For this reason, draws to the best possible hand are always way better than draws to strong, but potentially second-best hands.

When you are unsure if your draw is actually good if it arrives, you should discount some number of your outs, depending on the specific situation. For example, against one opponent with the four-high flush draw, it will usually be good if it completes, so perhaps you should assume you have eight outs instead of nine. However, if there is a bet and five callers before you, it is probably wise to fold your low flush draw immediately because it is highly likely someone else has a flush draw that is better than yours, meaning you could easily have zero outs.

As a shortcut to determine roughly how often your draw will complete, take the number of outs you have, multiply it by 2, and make it a percentage. This will let you know how often your draw will complete on the next card. For example, if you have an 8 out open-ended straight draw, it will arrive on the next card roughly 8 × 2 = 16% of the time. If you have a 15 out straight flush draw, it will arrive 15 × 2 = 30% of the time.

If there are two cards to come, assuming you will see them both (which may or may not be the case), you can multiply your number of outs by 4 and make it a percentage. For example, if you have an eight-out open-ended straight draw, it will arrive by the river roughly 8 × 4 = 32% of the time. If you have a 15 out straight flush draw, it will arrive roughly 15 × 4 = 60% of the time by the river.

Implied Odds

When you face a bet on the flop, you may or may not face another bet on the turn. For this reason, you cannot presume you will always get to see the river after calling a flop bet. However, if your opponent will frequently bet the turn after betting the flop regardless of what card comes on the turn, you may have large implied odds, which are additional odds you may realize when you improve to the best hand. This concept also applies when calling a turn bet because your opponent may bet again on the river.

Suppose you are playing $500 deep in a $2-$5 no-limit cash game. Your opponent raises to $15 before the flop and only you call from the big blind with 7 6. The flop comes 8 5 2 giving you an eight-out open-ended straight draw. You check and your opponent bets $30.

Your pot odds are 30/(32 + 30 + 30) = 33%. You know you will improve to a straight roughly 16% of the time on the turn. 16% is way less than 33%, so folding may seem ideal. However, your opponent could easily continue betting if the turn is a ten or five, meaning you are calling $30 now to potentially win $92 plus however much more your opponent will put into the pot on the turn and river when you improve.

This exact amount will be difficult to determine because your opponent will not automatically keep betting and put in all of their chips, but it is definitely some real amount that should be accounted for. Suppose you think your opponent will frequently bet $75 on the turn and will then bet $200 on the river 30% of the time (when they happen to have a really strong hand). This means you will get in perhaps $65 on average on the turn and perhaps $60 more on average on the river, meaning you have an additional $125 of implied odds.

You can add this additional $125 to the amount you can win on the flop to see if you are getting the correct pot odds plus implied odds to continue. You are now calling $30 to win the $32 pot, your opponent’s $30 bet, your $30 call, plus $125 implied odds, which is 30/(32 + 30 + 30 + 125) = 14%. Since you know you will improve on the turn roughly 16% of the time, and you know you only need to win 14% of the time to continue, you can call (raising is also an option).

It Is Not Actually That Simple…

Poker is a difficult game because there are many things you must consider, even in somewhat simple situations. In the previous situation where you have 7-6 on 8-5-2, consider what happens if the turn is a 7 or 6, giving you a pair. If your opponent bets again, you certainly cannot fold due to having more outs when you are behind and potentially even being ahead.

Another likely possibility is you could check-call your opponent’s flop bet, the turn checks through, and you fail to improve on the river. You may then have a profitable bluffing opportunity. Alternatively, you may hit your draw on the turn, check/call a bet, and then have a check-raise all-in on the river get called, winning your opponent’s stack.

So far, we have only discussed how to play your specific hand, but assuming your opponent plays well, you should actually consider how to play your entire range in a balanced manner, which massively complicates things. In the long run, you will have many different hands in this same flop situation, such as 9-9, A-8, 8-6, 7-7, 6-5, 4-4, A-2, Q-9, 10-9, 9-7, and 7-4, many of which should play in the same way as 7-6.

How to play in a balanced manner is beyond the scope of this article but is discussed extensively at PokerCoaching.com. Poker is a complex game, but if you break it down to its basic building blocks, it is possible to start developing reasonably sound strategies. If you consistently consider the basic math in this article, you will be well on your way to success.

If you want more resources to help you improve your game, I put together a course called Master the Fundamentals. This course covers the basics, preflop, post-flop, multiway, turn and river strategy, and much more. This course is completely free inside Card Player Poker School!

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Jonathan Little is a two-time WPT winner and the 2024 PokerGO Cup champion with nearly $9 million million in live tournament earnings, best-selling author of 15 educational poker books, and 2019 GPI Poker Personality of the Year. If you want to increase your poker skills and learn to crush the games, check out his training site at PokerCoaching.com/cardplayer.