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 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.
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