# 154 decision/high.or.low.p

## Description

This article is from the Puzzles FAQ,
by Chris Cole chris@questrel.questrel.com and Matthew Daly
mwdaly@pobox.com with numerous contributions by others.

# 154 decision/high.or.low.p

I pick two numbers, randomly, and tell you one of them. You are supposed

to guess whether this is the lower or higher one of the two numbers I

picked. Can you come up with a method of guessing that does better than

picking the response "low" or "high" randomly (i.e. probability to guess

right > .5) ?

decision/high.or.low.s

Pick any cumulative probability function P(x) such that a > b ==> P(a)

> P(b). Now if the number shown is y, guess "low" with probability

P(y) and "high" with probability 1-P(y). This strategy yields a

probability of > 1/2 of winning since the probability of being correct

is 1/2*( (1-P(a)) + P(b) ) = 1/2 + (P(b)-P(a)), which is > 1/2 by

assumption.

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