The probabilities of getting specific starting hands on preflop

The purpose of this article - is to explain the basic mathematics of Texas Hold’em on preflop. By the way, poker mathematics is very important aspect in the game, but at the same time it’s very simple, and does not require from you a deep knowledge in this area. In this specific article we will try to explain you how probabilities of getting specific starting hands on preflop work. And we will try to do it in a very simple language.

It’s extremely simple to calculate the probabilities of getting specific pocket cards on preflop. Every player is getting 2 starting hands in the beginning of a game (see rules of poker). Every player sees only his own cards and doesn’t see the cards of other players at the table. Thus, the first pocket card a player has been dealt can be one of 52 cards (because a deck of 52 cards is used in poker), and the second one - one of the 51 card left in the deck. Thus, the total number of possible card combinations in poker equals 52*51/2 = 1326.

The combinations in poker can also be estimated as a binomial coefficient: C(52,2)=1326. It’s extremely convenient to use the following formula:

C(n,k) = n! / (k! * (n - k)!)

Where "!" sign being a factorial.

From Wikipedia: A of integer ‘n’ is a product of all integers from 1 to ‘n’ inclusive. For example,5! = 1*2*3*4*5 = 120

In our example we get:

C(52, 2) = 52! / (2! * (52 - 2)!) = 52! / (2! * 50!)

Because 52! = 52 * 51 * 50!, then in the end we get:

The number of combinations of starting hands at preflop = C(n,k) = n! / (k! * (n - k)!) = C(52,2) = 52 * 51 * 50! / (2! * 50!) = 52 * 51 / 2! = 1326

Thus, we’ve learned how to determine the total number of starting hands at preflop and now are aware that there are 1326 of such hands. Please remember this number, it will help you in your further calculations.

Because there are no trump cards in poker, that is, all suits in the game are equal, the number of different combinations on preflop will be equal 169: 13 pocket pairs (6 combinations of each pair), 78 suited starting hands (4 combinations each) and 78 offsuit starting hands (12 combinations each). Let’s check: 13*6 + 78*4 + 78*12 = 78 + 312 + 936 = 1326. That’s right.

Also with our binomial coefficient, we can determine the probability to be dealt a pocket pair:

13 * C(4,2) / 1326 = 13 * 6 / 1326 = 78 / 1326 = 0.0588 or 17:1 (5,8%),

where C(4,2) will be a number of ways we can get a pair of four suits, for example for Kings -    ,    ,    ,    ,     and    .

Similar way we can determine the probability to get suited cards:

78 * С(4,1) / 1326 = 78 * 4 / 1326 = 0,2353 or 4,25:1 (23,53%)

As well as probability to get unpaired suited cards:

78 * С(4,1) * С(3,1) / 1326 = 936 / 1326 = 0,7059 или 0,417:1 (70,59%)

It turns out everything is pretty simple, isn’t it?

And finally, to clarify everything, we will present a chart with the odds to get different starting hands in Texas Hold’em:

Chart: Odds of getting different starting hands

If you would like to get a deeper knowledge of poker mathematics, please proceed to our Mathematics of poker section. Good luck!