Question

Assertion :A player tosses a coin and scores $1$ point for a head and $2$ points for a tail. He plays on until his score reaches or passes $n$. If $P_n$ denotes the probability of getting a score of exactly $n$, then $P_n+\frac{1}{2}{P}_{n-1}=P_2+\frac{1}{2}{P}_1.$ Reason: $P_n=\frac{1}{2}(P_{n-1}+P_{n-2}),n\ge 3$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$A$ score of $n$ can be reached in the following two mutuallly exclusive ways

- throwing a head when the score is $(n-1).$

- throwing a tail when the score is $(n-2).$

Hence, we have for $n\ge 3.$

$P_n=P_{n-1}P\left(H\right)+P_{n-2}P\left(T\right)$

$=\frac{1}{2}(P_{n-1}+P_{n-2})$ ...(1)

Now, from (1), we have

$P_n+\frac{1}{2}{P}_{n-1}=P_{n-1}+\frac{1}{2}{P}_{n-2}$

$=P_{n-2}+\frac{1}{2}{P}_{n-3}=...=P_2+\frac{1}{2}{P}_1$