It is tossed n times. Let `P_n` denote the probability that no two (or more) consecutive heads occur. Prove that `P_1 = 1,P_2 = 1 - p^2 and P_n= (1 - P) P_(n-1) + p(1 - P) P_(n-2)` for all `n leq 3`.

Answer

A

Explanation

Since, `p_(n)` denotes the probability that no two (or more) consecutive heads occur.
`impliesP_(n)` denotes the probability that 1 or no head occur. For `n=1, p_(1)=1` because in both cases we get less than two heads (H, T).
For `n=2,p_(2)=1-p` (two heads simultancously occur).
`=1-p(HH)=1-pp=1-p^(2)`
For `nge3,p_(n)=p_(n-1)(1-p)+p_(n-2)(1-p)p`
`impliesp_(n)=(1-p)p_(n-1)+p(1-p)p_(n)-2`
Hence proved