Mark  Poker Articles, Poker Mathematics
This article is a part of the Poker Mathematics series.
This article is number 1 in a planned series of 3 articles covering the (in my opinion) most important mathematical aspects of poker:
 Calculating pot odds: see Poker pot odds; all you need to know
 Calculating poker probabilities: see Poker probabilities; all you need to know
 Calculating expected value: see +EV poker; making the winning plays
I have a feeling that many poker players, both experienced and beginners, could do with a brushup on how to calculate poker pot odds. In this article I will try to explain the concepts so they make sense. Hope I succeed:)
Firstly when it comes to odds calculations there are basically 3 different systems:
 The Fractional Odds System favored among bookmakers in the UK. In the fractional odds system you will see odds displayed as fractions such as 6/1 or more commonly 6:1
 The European Decimal System used mostly by bookmakers in Europe (no surprises here). The European decimal system displays odds like decimal numbers such as 1,3, 2,7, 6 etc.
 The American Moneyline System favored by bookmakers in the US, hence the name of the system. In the American Moneyline System you will find odds like +500, 123, +345 etc.
I favor the European Decimal System (I’m European what can I say?) but a quick glance at the available poker literature out there will quickly convince you that the fractional odds system is favored among most poker writers (including my guru Dan Harrington). Luckily this article will give you an idea of how poker pot odds are calculated in all three systems.
Moving on the actual math: let’s say you are playing an online tournament and the pot after the flop is 2500. Your only remaining opponent bets 500, what are then your pot odds for calling? With your opponent’s bet of 500 the pot is now 3000 and you have to call 500 to stay in the pot. Here’s how you would calculate the pot odds for calling in each of the odds systems described about:
 In the Fractional Odds System the fraction quotes how much profit the bettor will make relative to his stake. A fractional odds of 3:1 therefore means that the bettor will make a 300$ profit for every 100$ staked. A fractional odds of 1:3 on the other hand means that the bettor will make a 33$ (100$*1/3) profit for every 100$ staked. In both cases the bettor will receive his original stake back. This means that in the 3:1 example the bettor will receive 400$ (300$+100$) in return and in the 1:3 example the bettor will receive 133$ (100$+33$) in return. Returning to the poker example from above, by calling your opponent’s bet of 500 you could potentially win a pot of 3000. A bet of 500 with a profit of 3000 gives a fractional pot odds of 6:1 (3000/500). As written earlier your initial stake is returned to you along with your profit which means that you in total will receive 3500.
 The European decimal system differs from the other two systems in as much as the bettor in effect hands over his stake to the bookmaker. The decimal odds then reflects the total amount that will be returned to the bettor; that is both his stake and his profit. Let’s see how this works in the 3:1 and 1:3 examples from above. When staking 100$ to win 300$ the bettor parts with his stake and is paid back his stake plus his profit which equals 400$ (100$+400$). In this case his decimal odds are quoted as 4 (400$/100$). In the 1:3 example, the decimal odds are 1,33 (133$/100$). Revisiting the poker pot odds example, the decimal poker pot odds are 7 (3500/500).
 In the American Moneyline system the odds are either positive or negative. The fractional odds of 3:1 from the example above would be written as +300 indicating the amount of money to be won on a 100$ wager. The 1:3 fractional odds would be written as 300 indicating that you would have to wager 300$ to win 100$. Going back to the poker pot odds example the 6:1 pot odds therefore corresponds to +600 in the American moneyline system. As a concluding remark I do not think the American moneyline system is well suited for calculating poker pot odds, so if I were you I would focus on getting familiar with either the fractional or the decimal odds system.
Given the explanations above, converting poker pot odds between the three systems is pretty straightforward. I have listed some examples below. Fill in the blanks yourself to practice converting odds.
 Fractional odds to decimal odds: write the fraction as its corresponding decimal number and then add one. Thus 8:1 corresponds to 9, 1:4 corresponds to 1,25, 1:8 corresponds to 1,125 and so on.
 Fractional odds to moneyline odds: if the odds is more than even (i.e the decimal odds is larger than 2), convert the fraction to its decimal number and multiply by 100. If the odds is less than even invert the fraction, convert to its decimal number and multiply by 100. For example 7:1 becomes +700, 2:1 becomes +200, 1/4 becomes 400, 1/9 becomes 900 and so on.
 Decimal odds to fractional odds:
 Decimal odds to moneyline odds:
 Moneyline odds to decimal odds:
 Moneyline odds to fractional odds:
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10 Comments to Poker pot odds; all you need to know
Great stuff Mark!! I have always considered whether pot odds was given in European or UK odds. Hence been concerned whether may calculations have been correct when evaluating whether to proceed in flush, straight draw situations.
October 27, 2008
Interesting stuff, especially the fact that you European’s do things very differently. LOL.
The main thing to keep in mind when calculating odds is that whatever system you use, use it to also calculate your card odds. It would confusing to calculate pot odds using fractional odds but then count card odds using decimal. For example: pot odds of 1:4 ($100 to win $400) while chasing a flush (9 outs with 47 cards left, assuming you have 2 in your hand and 2 on the flop) after the flop would give you 9:38 or approximately 1:4 and not 9:47 or approx. 1:5. When calculating your card odds or OUTS as many call it, you shouldn’t double count your cards. In the above example there are 9 flush cards left (your outs) in the deck (47 cards after 3 on the flop and 2 in your hand) meaning there are 38 cards that don’t help you. Now there could be more cards that help you (top pair, straight) but let’s keep this example as simple as possible to avoid too much confusion. So that leaves you with a 9:38 chance to hit your flush on the turn, or 1:4.22. Had you calculated it the other way you would be double counting your outs, once in our favour and once against you (9:47). I’ve seen people do this and let go of a hand because they thought their odds where not good, when in fact they were getting proper odds to make the call. Hope this helps, if not email me and we can discuss it more.
October 27, 2008
Hi Mctap
At first I thought you had misunderstood something when you started talking about 9/38 instead of 9/47. After doing some research I now realize you are right, however your comment added to my confusion:)
The probability of hitting your flush IS 9/47 however the fractional odds against an event are calculated according to the following formula:
Odds = (1p)/p, where p is the probability
In our example (1p)/p = (1(9/47))/(9/47) = (38/47)/(9/47) = 38/9 = 4,22/1
So the probability is 9/47, but the fractional odds are 38/9.
This is why I like using European decimal odds. Here the odds are simply calculated by taking the inverse of the probability.
Hence a probability of 9/47 corresponds to an odds against the event of 47/9 = 5,22.
I will attempt to clarify these issues in my next article.
October 27, 2008
After reading your response I still find myself confused with the terminology. The point I was trying to get at is that whatever system you use to calculate your pot odds, make sure you use the same to calculate your outs. The problem I often hear is that after calculating their pot odds (4:1 for a $100 bet to win $400) but then use probability to calculate their outs (9 outs with 47 cards left in the deck) instead of fractional (9 cards help and 38 don’t) and this doesn’t work out when deciding whether to call/raise or not. I know after reading my example again, I was a tad confusing in what I was trying to get across.
This just proves that this stuff is confusing at times.
October 28, 2008
I think the best is the European decimal system as you merely use a calculator to multiply the stakes with the actual odds.
October 28, 2008
Hi McTap
I hope to clarify the confusions in my next two articles. But you are right when you say that one should stick to the same odds system throughout the calculations.
I do not agree with you when you write “calculate your outs”. Outs are not calculated, they are specific cards that you can count. I think what you mean is that people should use the same odds system in order to calculate the correct expected value (EV) for the hand in question.
This is what I will go into more detail with in my next articles.
October 28, 2008
Great article and discission thread, one I think that highlights the fact that we all do things a little differently but what is important is that you get to the right answer. Personally, I use percentages… using your flush draw example I always calculate it like so, I have 9 outs, 9 outs after the flop means I am going to hit one of my cards 35% of the time (asssuming I see the turn and the river). I then calculate a few other things like positon in the tournament (I am not a cash player) what percentage of my stack I am risking etc. When calculating the odds of hitting a card that will improve your hand I think it’s handy to have these “outs” numbers at your fingertips, serisoulsy, if you play online you should print out that chart and pin it up beside your monitor. http://casinogambling.about.com/library/weekly/aa050103.htm
If you can’t remember a chart like that or don’t have a printer try to remember that the probability of any given card hitting after the flop is approximatley 2.1%. Multiply that by your number of outs and you have a pretty good estimate of what percentage of the time the next card will imrpove your hand, of course you do the math all over again after the turn and before the river.
October 28, 2008
Hi JGiles
Thanks for your input! I will elaborate on your “easy to remember rule of thumb” in my next article.
Best Regards
Mark
July 14, 2010
Cheers to the clearcut explanation, computation and details. I agree, some poker players, whether just starting or are pros already sometimes need an overview on calculating pots. Good job you’ve done here!
July 16, 2010
Cheers to the clearcut explanation, computation and details. I agree, some poker players, whether just starting or are pros already sometimes need an overview on calculating pots. Good job you’ve done here!
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