That pesky A of H

tivol at tivol at
Wed Jun 15 15:59:42 EST 1994

Sean Eddy still uses Bayes' rule.  However...
The total number of west hands is 26!/(13!*13!), i.e. the number of ways the
26 missing cards can be taken 13 at a time.  The number of hands with the A
of D in the west is 25!/(13!*12!).  The A of H can be in one of the 25 pos-
itions not occupied by the A of D, so the number of hands with both red aces
in the west is (12/25)*[25!/(13!*12!)] or 24!/(13!*11!), the number of hands
with the A of D in west and the A of H in east is 24!/(12!*12!).  No other
possibilities need be considered.  The number of hands where a red ace is the
first card (equivalent to seeing one card) is [24!/(13!*11!)]*[2/13] for the
case where west has both red aces, and [24!/(12!*12!)]*[1/13] for the case
where west has the A of D, but not the A of H.  The sum of these two is the
total number of hands which match the conditions of the problem, and it is
[24!/(12!*11!)]*[2/(13*13)+1/(13*12)].  The ratio of this to the number of
hands where west has both aces one of which is seen is [2/(13*13)]/[2/(13*13)
+1/(13*12), or 2/[2+(13/12)]  Oops, I meant the ratio of both/one seen to the
total... Anyway, the probability is 24/37; closer to Sean's answer than mine.
One further consideration for the case that the trick is complete.  Bradley
Sherman doesn't have things quite right.  If east *follows* to the diamond
lead, the odds are not changed; however, if east sluffs on the diamond lead
then (assuming north-south do not have the remaining 12 diamonds in their
hands) the odds are changed--sometimes considerably, as when west holds all
13 diamonds!  The odds also change dramatically if east sluffs the A of H on
the diamond lead!  So, if east has not played to the first trick or if north
and south have all the remaining diamonds, Sean's Bayesian analysis holds if
the exact probabilities are used in the formula, and if east sluffs on the 
first trick and north and south do not have all the remaining diamonds, a
more thorough analysis is required.


				Bill Tivol

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