Suppose you are heads up in a tournament and that you have t

Suppose you are heads up in a tournament, and that you have two chips left and your opponent has four chips left. Suppose also that the results on different hands are iid, and that on each hand with probability p, you gain one chip from your opponent, and with probability q, your opponent gains one chip from you.

(a) If p = 0.52, find the probability that you will win the tournament.

(b) What would p need to be so that the probability that you will win the tournament is 1/2?

(c) If p = 0.75 and your opponent has ten chips left instead of four what is the probability that you will win the tournament? What if your opponent has 1,000 chips left?

Solution

p for gaining and q for losing

In the beginning you have 2 chips and your opponent 4 chips.

If you get all the 6 chips and again win, you win as a whole

HEnce Prob (winning) = prob of winning 5 more times consecutively

= (0.52)⁵

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If prob of winning = p⁵ = 1/2

Then p = 5th root of 0.5 = 0.8706

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c) if p =0.75 and opponent has 10 chips left, consecutively 11 times should be won

prob = (0.75)¹¹

For 1000 chips prob of winning = (0.75)¹⁰⁰¹