Consider the archery competition in Example 6.

(a) Assume that Nina goes first. Let pn represent the probability that Brook wins on his nth turn. Give an expression for pn. 

(b) Use the result from (a) and a geometric series to determine the probability that Brook wins when Nina goes first.
(c) Now assume that Brook goes first. Use a geometric series to compute the probability that Brook wins the competition.

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EXAMPLE 6 A Probability Computation Nina and Brook are participating in an archery competition where they take turns shooting at a target. The first one to hit the bullseye wins. Nina's success rate hitting the bullseye is 45%, while Brook's is 52%. Nina pointed out this difference, arguing that she should go first. Brook agreed to give the first turn to Nina. Should he have?