Poker Math

Ok. If you are dealt 6 cards. What are the odds to get a 6 card flush from the first 6 cards? I think I got that. But. If you can draw 3 cards after getting your first 6 card hand what are the odds to get a 6 card flush. I'm not sure how the math works when you can draw 3 cards.

For getting dealt a 6 card flush out of 6 cards, assuming a regular 52 card deck, the probability is:

(52/52) * (12/51) * (11/50) * (10/49) * (9/48) * (8/47) = 0.00033715613 or 1-in-2965.98484848.

That would include straight flushes as well as royal flushes.



When drawing, if you draw in such a way to maximize the probability of getting a flush (ie: you're dealt AAAAKQ and you hold 2 suited cards), then the math works out similarly, but you have to take into account the likelihood of each starting hand. Since there's no strategy and all you do is hold as many suited cards that you have, it'll work out to this:

1. What is the probability you're dealt only 2 suited cards? Figure out that probability, then multiply it by (11/46)*(10/45)*(9/44)*(8/43).

Add that number to the following:

2. What is the probability of only getting dealt 3 suited cards? Multiply that by (10/46)*(9/45)*(8/44).

Add that number to the following:

3. Same thing, but with 4 suited cards....

4. Same thing, but 5 suited cards.



Add all those numbers up, and there you go.