Re: 2 Royal flushes in 1000 hands
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The probability of seeing a royal flush after 7 cards is
(4 * 47 * 46 )/ (52 choose 7) = ~0.00006464124 or about 1 in 15 470
To find the chance of this happening exactly twice in 1000 hands, use EXCEL BINOMDIST (2,1000,0.00006464,false) = 0.001956684
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That needs to be P = (4*47*46/2)/C(52,7) = 4*C(47,2)/C(52,7) =~ 0.00003232062
or 1 in 30,940. Then the probability of exactly 2 in 1000 is C(1000,2)*P^2*(1-P)^998 or BINOMDIST(2,1000,P,FALSE) =~ 0.000505226 or 1 in 1,979.
If you must use both hole cards, then the probability of a royal is only P = 4*C(5,2)*C(47,2)/C(52,2)/C(47,5) =~ 0.00002125857 or 1 in 47,040. The probability of exactly 2 in 1000 is C(1000,2)*P^2*(1-P)^998 or BINOMDIST(2,1000,P,FALSE) =~ 0.0002209986 or 1 in 4,525.
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