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\line{\bf The Plausible Improbable Scenario\hfil}

On his way to the quarterfinals in the 1984 US Amateur Championship, my
friend and sometime doubles partner Sam Hanna ($X$) was leading
Randy Prater ($O$) by~5 to~1 in a seven point match in the position below.
Sam declined to double, fearing that Prater would take, redouble, hit
a blot, and equalize the score. Sam should have doubled and prayed for a take.

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\noindent
If Prater drops a double, experts estmate that he has about a 9\% chance
in the 6--1 match against an equal opponent. To take and win the game requires
a long parlay of favorable events:

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Sam leaves a shot 44\% of the time.

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Randy hits it 33\% of the time.

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Randy contains the checker and wins the bearoff half the time.

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Randy wins the resulting tied match half the time.

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So his chance is $0.44\times 0.33\times 0.5\times 0.5=0.037$, a bit under 4\%.
He must drop a double.

Sam saw that he had a plausible, and therefore scary, way to lose the match.
He forgot for a moment what he tells his Boston University classes about
probability:

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The probability of a parlay is the product of the probabilities of its stages.

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If each stage has a probability no more than 50\%, the probability of 
a two-stage parlay is no more than 25\%; three stages, 12\%; four stages, 6\%;
five stages, 3\%.

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Sam's scenario for losing seemed plausible, because every step has a
substantial probability, yet the product law makes it hopelessly improbable.


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\copyright 1987 Robert W. Floyd

First draft August 14, 1987

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