2009 AIME I Problems/Problem 3
Revision as of 19:25, 19 March 2009 by Projectstartrek (talk | contribs) (New page: == Problem == A coin that comes up heads with probability and tails with probability independently on each flip is flipped eight times. Suppose the probability of three heads and five t...)
Problem
A coin that comes up heads with probability and tails with probability independently on each flip is flipped eight times. Suppose the probability of three heads and five tails is equal to of the probability of five heads and three tails. Let , where and are relatively prime positive integers. Find .
Solution
If we let the odds of a tails (1-p) equal t, then the probablity of three heads and five tails is: p^3t^5 The probability of five heads and three tails is: p^5t^3
25p^3t^5 = p^5t^3 25t^2 = p^2 5t = p 5(1-p) = p 5 - 5p = p 5 = 6p p = 5/6
