Re: Another Tack on the Royal Flush
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You are observing a 7-handed game of texas hold'em. What is the probability that by the river someone (or everyone) would have a royal flush, if they stayed in the hand?
In other words, if nobody folds and all seven players show down, what is the probability that at least one player will have a royal flush?
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For both 3 and 4 parts on the board, only 1 player can have it, so the only way that more than 1 player can make a royal is if the board is a royal, and then all 7 players have it. The probability of making a royal with any 7 cards is 4*C(47,2)/C(52,7). If we multiply this by 7, it will count the cases where there is a royal on the board 7 times, so we then subtract off 6 times the probability that the board is a royal, which is 6*4/C(52,5).
7*4*C(47,2)/C(52,7) - 6*4/(52,5)
=~ 0.0217% =~ 1 in 4608.
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