Probabilities

The following calculates the frequency of each poker hand for all combinations that can be made out of 5 cards drawn at random from a full deck of 52 playing cards, without the use of wild cards.

The probability in poker is determined based on the number 2,598,960, which represents the total number of five card combinations that can be created. In this case, the probability is the frequency of a hand divided by the total number of five card hands, and the odds are defined by the formula ((1 / p) – 1 : 1), where p corresponds to the probability.

Frequency of Five Card Poker Hands

In the following table, the frequencies are precise, while the probabilities and the odds given are approximate.

Also, the royal flush is included in the table as a straight flush, since the royal flush can be made in four different ways (hearts, spades, clubs and diamonds). This calculates a probability of 0.000001539077169 and odds of 649,739 : 1.

Frequency of 5 Card Poker Hands
HandFrequencyProbabilityOdds against
Straight flush400.00154 %64,973 : 1
Four of a kind6240.0240 %4,164 : 1
Full house3,7440.144 %693 : 1
Flush5,1080.197 %508 : 1
Straight10,2000.392 %254 : 1
Three of a kind54,9122.11 %46.3 : 1
Two pair123,5524.75 %20.0 : 1
One pair1,098,24042.3 %1.37 : 1
No pair1,302,54050.1 %0.995 : 1
Total2,598,960100 %0 : 1

The probability of a straight flush or an ace low straight will be reduced in case they won’t be counted. They will become 9/10 as common as they usually would be.

Pot Odds

In poker, pot odds are used to determine the expected value of a play. Poker players use pot odds to calculate the profitability over the long term. Generally, odds are expressed as a win:loss ratio. The following formula can be used to calculate the percentage probability:

win:loss odds = win / (win + loss) % probability

For instance, this means that 1:4 odds equals 1 / (1 + 4) = 20% probability.

Instead, odds are also expressed as a loss:win ratio, or also known as odds against (see table above). To make it totally clear, we have used win:loss odds in remainder of this explanation.

The correct poker playing strategy is influenced by pot odds for every possible table action (check, fold, call or raise) at every single moment during a poker game.

Most likely, calling is the correct play when facing a call with higher pot odds, just like folding is the correct play when facing a call with lower pot odds.

For instance, if you can win a $1000 pot by making a call for just $1, essentially no hand should be folded, since you only have to win this call one time every thousand hands for it to be profitable.

The probability of winning is equal to the chance that a player will win with either the best hand at showdown or simply because the opponents will fold out of the hand before opening up.

Pot Odds Texas Hold’em Poker

The approximate percentage probability that a player hits one of his outs on the first coming card in Texas Hold’em is calculated as follows:

(number of outs) x 2 + 1

In case you have the outs for a potential flush, i.e. 9 cards could improve your hand, then you’ll have about 19% chance to complete your flush with the next card (9 x 2 + 1 = 19).

With two more cards to come (the turn and the river), the approximate percentage probability is calculated as follows:

(number of outs) x 4 – 1

This means that you’ll have 35% roughly to hit a flush with two more cards to come (9 x 4 – 1 = 35).

For a correctly classified table action, the odds for winning a hand should never be lower than the applicable pot odds.