Question

30.Consider a coin that comes up heads with probability p and tails with probability 1 − p. Let qn be the probability that after n independent tosses, there have been an even number of heads. Derive a recursion that relates qn to qn−1, and solve this recursion to establish the formula qn = 1 + (1 − 2p) n 2

Using method other than Mathematical Induction

Answer 1

Answer:-

Given That:-

Consider a coin that comes up heads with probability p and tails with probability 1 − p. Let qn be the probability that after n independent tosses, there have been an even number of heads. Derive a recursion that relates qn to qn−1, and solve this recursion to establish the formula qn = 1 + (1 − 2p)n /2.

Given,

= P[even number of heads in n independent tosses]

= P[odd number of heads in (n - 1) independent tosses and nth toss is head] + P[even number of heads in (n - 1) independent tosses and nth toss is tail]

  

-

-

-

-

-

-

Proved.

Thank you for your supporting. please upvote my answer...