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Complementary Event

The value of ...

Question

The value of $P_{n}$ is equal to

(1 / 2) [P_{n - 1} + P_{n - 2}]

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(1 / 2) [2 P_{n - 1} + P_{n - 2}]

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(1 / 2) [P_{n - 1} - P_{n - 2}]

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n o n e o f t h e s e

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Solution

The correct option is A $(1 / 2) [P_{n - 1} + P_{n - 2}]$
We can score exactly n in two ways: by scoring exactly n−1, and then throwing a head; or by not scoring n-1.
Let Pn be the probability that our random walk passes through
n. Since the first step is 1 or 2, each with probabiliyt 1/2
So , $P_{n + 1}$ =1/2 $P_{n}$ +1/2 $P_{n - 1}$
With inital condition $P_{0}$ =1, $P_{1}$ =1/2

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