Take the 2-minute tour ×
Poker Stack Exchange is a question and answer site for serious players and enthusiasts of poker. It's 100% free, no registration required.

The following is a passage from Wikipedia on starting hands probability:

The 1,326 starting hands can be reduced for purposes of determining the probability of starting hands for Hold 'em—since suits have no relative value in poker, many of these hands are identical in value before the flop. The only factors determining the strength of a starting hand are the ranks of the cards and whether the cards share the same suit. Of the 1,326 combinations, there are 169 distinct starting hands grouped into three shapes: 13 pocket pairs (paired hole cards), 13 × 12 ÷ 2 = 78 suited hands and 78 unsuited hands; 13 + 78 + 78 = 169. The relative probability of being dealt a hand of each given shape is different. The following shows the probabilities and odds of being dealt each type of starting hand.

enter image description here

The equation used to calculate the total possible number of starting suited hands and unsuited hands is 13 x 12 ÷ 2, but what I don't get and try to figure out is what does 12 and 2 represent in the equation?

Another thing I'm confused about is how the suit combinations for each unsuited non-paired hand is calculated?

share|improve this question
add comment

1 Answer

When you are dealt a card, it has 13 possible ranks. When you are dealt a second card, for it not to be a pocket pair, that can be of any rank different to your first hole card - i.e. 12 different card ranks. For the division by 2, the calculations are for the number of combinations of different types of hands and a combination disregards order. If you take 13 * 12 you get each possible non-paired hand twice, e.g., AK and KA or 74 and 47, so you divide by 2 to count each combo only once. Thus the equation 13 × 12 ÷ 2 = 78.

For the 12 possible (suit combinations for each hand) for (unsuited cards nonpaired) hand shape equation, the 4 represents the number of suits and the 3 represents the number of suits any single suit can be paired with to give an offsuit combination. Four suits x three combinations each = 12 combinations total. The equation is not a fraction, the 1 in the equations is not a denominator. The symbol with brackets such as (4 1) is for the number of combinations of 4 things taken 1 at a time.

share|improve this answer
    
Just to add additional info, as you mention, "Combinations" DON'T consider the relative order of items, whereas "Permutations" DO consider the relative order of items. –  Toby Booth Mar 11 '13 at 22:52
    
@TobyBooth - With combinations, order doesn't matter, with permutations, order does matter. But surely you know this. Are you using the word "consider" in a way I don't understand? –  TTT Mar 12 '13 at 17:24
    
@TTT Oops, Typo. Just mixed them up ;) Thanks for catching that! Fixing now. –  Toby Booth Mar 12 '13 at 19:02
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.