Robb's 3000th Post

Hand rankings are based on taking 5 random cards out of a 52 card deck. We assign certain combinations of numbers and suits to be scoring hands. We then determine the frequency of drawing those scoring hands, which determines the rank of the hands.

How do the probabilities change when we're looking at the best 5-card hand from any 7 random cards?

I.e. There is ~51% chance of having a 'high card' hand with 5 random cards, what is the chance of having less than a pair with 7 cards?

Originally Posted by MadMojoMonkey

How do the probabilities change when we're looking at the best 5-card hand from any 7 random cards?

This is harder, as "weird" hands exist: 3 pair, double trips, 6-flushes, 7-flushes, straight + pairs (trips), flushes + pairs (trips), quads + trips. Counting them isn't any different, it just takes a while to think of all the oddities and figuring out which "normal" categories they fit into.

For one pair hands, we'd like to do something simple like:

P = 13c1 * 4c2 * 12c5 * (4c1)^5 / 52c7 ~ 47.28%

This gives a pair and 5 other card values. The problem is that several straights and flushes are in that figure, and it takes a while to count them and subtract them out.

Also, there are 133 million 7-card poker hands, almost a 2 order of magnitude increase in the size (and difficulty) of the problem.

I'll give it a shot in the morning, but it's easier to just freakin' google it.