A probability question
Apr. 22nd, 2006 06:04 pmI've attempted to formalise mathemetically the most interesting question in http://cartesiandaemon.livejournal.com/179726.html. It should have a solution now, I'm not sure if a trivial one or an unfindable one.
Consider Pa and Po in [0,1]. Have a series of trials (independent random variables taking values 0 or 1) X1,X2,..., succeeding with probabilities x1,x2,..., and an event H that the trials are in the subset of possible results S∈{0,1}N.
(We should add the constraint that being in S is determined by some finite initial segment of the sequence, or is so almost certainly, but can do so later if necessary.)
Each xi must be either (a) equal to Po, or (b) a function of Pa (with values in [0,1]).
Question: Can we choose x1,x2... and S thusly such that for any Pa and Po:
P(H)=Po if Pa=Po
P(H)<Po if Pa>Po
?
Consider Pa and Po in [0,1]. Have a series of trials (independent random variables taking values 0 or 1) X1,X2,..., succeeding with probabilities x1,x2,..., and an event H that the trials are in the subset of possible results S∈{0,1}N.
(We should add the constraint that being in S is determined by some finite initial segment of the sequence, or is so almost certainly, but can do so later if necessary.)
Each xi must be either (a) equal to Po, or (b) a function of Pa (with values in [0,1]).
Question: Can we choose x1,x2... and S thusly such that for any Pa and Po:
P(H)=Po if Pa=Po
P(H)<Po if Pa>Po
?