Poker Probability - Part 3: Calculating Holdem Probabilities PokerLaws.Org

Probability is one of the most important poker strategy concepts, but calculations can be complex; so complex, in fact, that we aren’t always able to do them on the fly in a game. That’s why it’s useful to know some common poker probabilities from memory.

It’s easy to look at a chart of common probabilities, but I’ve found that doing so rarely helps the information to stick. In order to really get a feel for the odds in poker, we need to perform the calculations ourselves at some point. So in this article, we’ll learn how to calculate some important poker probabilities – step by step.

Being Dealt A Specific Pocket Pair Preflop

The probability of being dealt a particular pocket pair preflop is easy to calculate. Just find the conditional probability of the following two events:

A. First card is of rank X
B. Second card is of rank X

We’ll use pocket aces as an example. The probability of event A is just the number of aces in the deck divided by the number of possible cards:

P(A) = 4/52

The probability of event B is the number of remaining aces divided by the number of remaining possible cards:

P(B | A) = 3/51

Finally, the conditional probability of events A and B occurring is:

P(Pocket Aces) = P(A) * P(B | A)
P(Pocket Aces) = 4/52 * 3/51
P(Pocket Aces) = 1/221

So the probability of being dealt pocket Aces preflop is 1/22. The cool thing here is that this probability applies not only to pocket Aces, but any pocket pair. So the probability of getting twos preflop is 1/221; the probability of getting tens preflop is 1/221; and so on.

Being Dealt Any Pocket Pair Preflop

Knowing the probability of getting a specific pocket pair preflop helps us to calculate the probability of getting any pocket pair preflop. We just need to sum the probabilities of getting a specific pocket pair for every possible pocket pair.

There are 13 different ranks in a deck of cards, which means there are 13 possible pocket pairs. Summing the probabilities for each of the 13 pocket pairs, we get:

P(Any Pocket Pair) = 13 * (1/221)
P(Any Pocket Pair) = 13/221 = 1/17

So we can expect to be dealt a pocket pair about once every 17 hands in Holdem.

Hitting A Set on the Flop

We can find out how often our pocket pairs will hit a set on the flop with a few simple calculations. We just need to calculate the probability that:

A. The first flop card will match our hand’s rank, OR
B. The second flop card will match our hand’s rank, OR
C. The third flop card will match our hand’s rank.

The “or” in the above descriptions makes it seem like these are mutually exclusive events. Given this, we might try to sum their probabilities to obtain our answer. However it turns out that these events are non-mutually exclusive. Thus we need to multiply instead.

We could attempt to calculate this Holdem probability by multiplying the probability of each flop card’s matching our pair’s rank. However this would result in double-counting some situations. To obtain an accurate number, we should calculate the probability of each flop card not matching our pair’s rank instead. Then we subtract that probability from 1 to obtain its complement, which will be the actual probability of flopping a set.

For the first flop card, there are 48 cards that miss our pair from 50 total cards in the deck. Thus

P(A) = 48/50

For the second flop card, there are 47 cards that miss our pair from 49 remaining cards in the deck. So

P(B | A) = 47/49

And for the third flop card, there are 46 non-useful cards from 48 remaining. This gives us

P(C | A and B) = 46/48

To find out the probability of not hitting a set on the flop, we take

P(A and B and C) = P(A and B) * P(C | P(A and B))
P(A and B and C) = (48/50 * 47/49) * 46/48
P(A and B and C) = 0.88244898

If we take the complement of P(A and B and C), we obtain the probability of hitting a set on the flop:

P(flop a set) = complement(P(A and B and C))
P(flop a set) = 1 – P(A and B and C)
P(flop a set) = 1 – 0.88244898
P(flop a set) = 0.11755

Which is roughly 11.75%. So when dealt a pocket pair, we can expect to hit a set 11.75 times out of 100.

Hitting a Four-Flush From Turn to River

Often, we’ll be in drawing situations where we haven’t quite made a flush. In these cases, it’s useful to know how likely we are to complete our flush.

Say we have a four-flush (four suited cards) on the turn. How likely are we to complete the flush on the river? Calculating this Holdem probability is dead-easy: we just take the probability that the river falls our suit.

We know that if we have four cards of a particular suit, there must be nine left in the deck. We also know that there are forty-six unseen cards in the deck on the turn. Thus the probability of hitting our flush on the river is simply

P(hit four-flush on river) = 9/46

Which translates roughly to 19.57%.

Hitting a Four-Flush From Flop to River

Say we’ve flopped a four-flush, and want to know how likely it is that we’ll complete by the river. Well, we already know the probability of completing from turn to river: 19.57%. Can’t we just multiply that by two to account for the extra street?

No, because we’ll end up double-counting and overestimating our chances – in this case by over 5%! We ran into this problem when we calculated the probability of flopping a set. We can avoid it here the same way we did there: by calculating the probability of not hitting a card of our suit on turn or river, and then taking its complement.

On the turn there are 47 unseen cards left in the deck, 38 of which don’t help us (47 minus the 9 remaining of our suit). On the river there are 46 unseen cards left, 37 of which don’t help us. This gives:

P(miss flush) = 38/47 * 37/46
P(miss flush) = 0.65

Which translates nicely to 65%. To find out the probability of hitting our flush by the river, we just take the complement:

P(hit flush) = 1 – P(miss flush)
P(hit flush) = 1 – 0.65
P(hit flush) = 0.35

Or 35%. This means we expect to complete a four-flush by the river 35 out of 100 times.

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