Strategy for an Unfair game

Probability Level 3

I n s p i r a t i o n : \mathbf {Inspiration:} A and B play a series of games where the probability of winning p \mathit p for A is kept less than 1 2 \frac{1}{2} . However A gets to choose in advance the total no. of plays. To win the game one must score more than half the games . If the total no. of games is to be even say 2 n 2n then choose the one that applies

Details and Assumptions: All you need to find is an optimal number of games that A should play so that his total probability of winning is maximized; If the probability of winning each play is p p then if p < < 0.5 p<<0.5 then it readily follows that the number of games A must play is 2; but if p p is near about but less than 0.5 0.5 then A might find a strategy to increase his total probability of winning. That's what is asked for in the question

2 1 − 2 p − 1 ≤ 2 n ≤ 2 1 − 2 p + 1 \frac{2}{1-2p}-1 \leq 2n \leq \frac{2}{1-2p}+1 1 1 − 2 ( 1 − p ) − 1 ≤ 2 n ≤ 1 1 − 2 ( 1 − p ) + 1 \frac{1}{1-2(1-p)}-1 \leq 2n \leq \frac{1}{1-2(1-p)}+1 1 1 − 2 p − 1 ≤ 2 n ≤ 1 1 − 2 p + 1 \frac{1}{1-2p}-1 \leq 2n \leq \frac{1}{1-2p}+1 1 1 − 2 ( 1 + p ) − 1 ≤ 2 n ≤ 1 1 − 2 ( 1 + p ) + 1 \frac{1}{1-2(1+p)}-1 \leq 2n \leq \frac{1}{1-2(1+p)}+1

Ariijit Dey by Ariijit Dey , 22, India

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