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Old 09-17-2007, 09:24 AM
bigpooch bigpooch is offline
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Join Date: Sep 2003
Location: Hong Kong
Posts: 1,330
Default Re: Five card draw card removal effects (LONG reply)

Actually, these are estimates; exact numbers would be ratios
over C(47,5) = 1533939 and the relevant combinations are in
my previous post from which exact numbers can be obtained
when heads up.

These approximations seem correct except that it is
misleading for the ties to be exactly the same hand with the
same kickers because in practice, JJT42 and JJ943 have the
same pot equity when these hands are in (with the intention
of drawing three to a pair). Hence, the ratio of
11559/C(47,5) or about 0.007535501738 should be thought of
and not 27/C(47,5) or about 0.000017601743.

Also, once we have numbers for heads up, an approximation
should be used when considering more than two opponents
because the inclusion-exclusion calculation is horrendously
difficult for an exact probability that the hand is best
predraw; besides, because of the factor in the next
paragraph, it is sufficient to have a good approximation.

In addition, if there has already been some action, such as
one or two players have folded ahead of you, the ranks are
no longer equally distributed but rather very slightly
skewed towards kings and aces.
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