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TwoBucks
11-20-2003, 02:42 PM
Could someone please tell me what the odds are of having a pair of jacks or better when dealt SIX cards (using the best 5 to make the hand of course). Like a video poker game that deals 6 cards instead of 5. THANK YOU!!

bigpooch
11-20-2003, 08:22 PM
It's best to look at the chances of getting a hand that
is worse than two Jacks and subtracting that from 1.
There are C(52,6)= 20358520 possible hands. The number
of nonpairs is then (C(13,6)-(9+56))x(4**6-(4+4x6x3)) =
6637020. The first multiplicand is the number of
nonstraighting combinations of ranks and the other
factor is the number of nonflush suit combinations.

For pairs, let's consider a pair of deuces:
there are 6x((C(12,4)-2)(4**4-2)) combinations; for a
pair of threes, C(12,4)-3 is used above instead and for
fours C(12,4)-4 is used and for pairs of fives to tens
C(12,4)-5 is used (these are the possible side card
combinations of ranks excluding straights). Summing,
there are then 6x(9xC(12,4)-39)(4**4-2) = 6729984
combinations of pairs of tens and less.

The sum of the two numbers above is 13367004 and thus
the number of hands which include all hands JJ and
stronger is then 20358520 - 13367004 = 6991516 which
gives the chances at about 0.343419659 of getting a hand
that is JJ or better.