A player tosses an unbiased coin and is to score two points for every head turned up and one poi...

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A player tosses an unbiased coin and is to score two points for every head turned up and one point for every tail turned up. If \( P_{n} \) denotes the probability that his score is exactly \( n \) points, prove that
\[
P_{n}-P_{n-1}=\frac{1}{2}\left(P_{n-2}-P_{n-1}\right) \quad n \geq 3
\]
Also compute \( P_{1} \) and \( P_{2} \) and hence deduce the probability that he scores exactly 4 .
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