Question

Bob and Alice are taking turn shooting arrows at a target, Bob being the first one to shoot. Alice hits the center of the target with probability r while Bob hits it with probability p. The first person to hit the center of the target wins.

(a) What is the probability that Alice wins

(b) Let X be the number of times that Alice shoots an arrow in the game. Find the PMF of X.

(c) For all k, find the conditional probability P[X = k|Alice wins]. Is it a known distribution?

Answer 1

Answer:

a)

Let Alice hits the target at ith try, given that all previous tries were unsuccessful by Bob abd Alice.

Probability Alice wins =

(Using summation of infinite Geometric Series)

b)

The PMF of X is,

for i = 1, 2, ...

c)

P(X = k given Alice wins) =  P(X = k and Alice wins) / P(Alice wins)

= P(X = k) / P(Alice wins)

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