Mark  Poker Articles, Poker Mathematics
This article is a part of the Poker Mathematics series.
This article is number 2 in a 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
Calculating exact poker probabilities for each hand you play can be a complex mathematical operation which is simply not feasible with the short response times in online, and to a lesser extent live poker. Poker players have different approaches to deal with this challenge. Some players use poker tools in the form of programs that continuously calculate and display probabilities and pot odds. Others memorize the probabilities of a wide range of poker situations. There are players who don’t calculate exact probabilities but instead base their decisions on previous experiences and intuition (I myself belong to this group, but am motivated to include more mathematical considerations in my game). In this article I will explain how exact poker probabilities are calculated from the number of outs you have and also present a very useful shortcut to calculating poker probabilities from outs.
First of all a clear definition of a poker out is needed: If you believe your hand needs improving after the flop to win the pot, then an out is a card that will do just that. Say for example that you are holding AK suited and the flop gives you two of your suit. Your opponent has QQ (none in your suit), so you need to improve your hand in order to win the pot. Your outs would be any card of your suit (9 remaining in the deck) or any A or K (6 remaining in the deck) giving a total of 15 outs. Keep in mind that counting your outs is always an estimate. If your opponent has flopped a monster what you might consider outs will end up costing you your stack. On the other hand your opponent could also be bluffing in which case your AK might already be the best hand.
Once you have estimated how many outs you have, calculating the probability of either the turn or the river being one of your outs is fairly straightforward. Simply divide the number of outs with the remaining number of unseen cards in the deck. Using the example from above the probability of the turn (P(turn)) being one of your suit, an Ace or a King is 15 divided by 47 (52 card minus your two hole cards and the three cards constituting the flop) or roughly 32%. If one of your outs did not come on the turn and the board did not improve your opponents QQ you still have 15 outs on the river giving you a probability of 15/46 or roughly 33% (P(river)) that one of your outs will come on the river.
Calculating the probability that one of your outs from the example above will come on either the turn or the river (P(turn or river)) is a bit more tricky. This probability is NOT simply the sum of P(turn) and P(river) because by summing these two probabilities you will be including outcomes that are not possible. For example if the Ace of Clubs hits on the turn it cannot come on the river. In mathematical terms the events of one of your outs coming on the turn or the river are not independent. I will give you two ways of calculating the exact probability of one of your outs coming on either the turn or the river.
Method 1:
There are 3 possible outcomes of hitting on of your outs on the turn or the river, namely you hit one on the turn and not on the river (P(turn not river)), you don’t hit one on the turn but one comes on the river (P(river not turn)) and you hit one on both the turn and the river (P(turn and river)). The probabilities are calculated as follows:
P(turn not river)= P(an out comes on the turn)*P(an out does not come on the river after an out has come on the turn) = (15/47) * (4614)/46 = 0,222
P(river not turn)= P(an out comes on the river)*P(an out does not come on the turn) = (15/46) * (4715)/47 = 0,222
P(turn and river)= P(an out comes on the turn)*P(an out comes on the river after an out has come on the turn) = (15/47)*(14/46) = 0,0971
Summing the 3 probabilities from above gives 0,54 which is the probability that one of your 15 outs will hit on either the turn or the river
Method 2:
The probability of hitting one of your 15 outs on either the turn or the river is the complement of the probability not hitting one of your outs on both the turn and the river:
P(turn or river) = 1 P(an out does not come on the turn)*P(an out does not come on the river) = 1((4715)/47))*((4615)/46)) = 0,54
I hope you agree with me that none of the methods above are very practical when it comes to calculating poker probabilities when you are sitting at the poker tables. Luckily there is an easy to remember rule of thumb that does a good and quick job of calculating poker probabilities from poker outs.
Easy rule of thumb:
P(turn or river) = (4*(number of outs) – 1)%
P(turn) = (2*(number of outs) +1)%
P(river) = (2*(number of outs) +1)%
Taking our 15 out example from above, the easy rule of thumb yields the following probabilities:
P(turn or river) = 59% = 0,59
P(turn) = 31% = 0,31
P(river) = 31% = 0,31
which is accurate enough for most purposes.
I will leave it to you to practise estimating how many outs you have in any given poker hand since this is one the most important disciplines in poker. The list below should get you started:
Outs when you are drawing to:
 Three of a kind: 2
 High Pair: 3
 Full house when you have hit two pair: 4
 Openended straight: 8
 Flush: 9
 Flush or high pair: 12
 Flush or pair: 15
 Openended straight, flush or pair: 21
In my next article I will explain how to use the concepts of poker pot odds and poker probabilities to become a winning poker player in the long run.
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5 Comments to Poker probabilities; all you need to know
Good stuff!
I had no idea that this rule of thumb existed. Makes it quite easy to calculate the correct probabilities.
Keep up the good work!
November 2, 2008
I can see I still have a lot to learn about poker probabilities.
Thanks for the info!
November 3, 2008
Great article .
February 9, 2009
Cool article, I’m going to use this when I play poker against my friends. This article may just help me win a hundred bucks.
February 9, 2009
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November 2, 2008