Question

A person whose hobby is tossing a fair coin is to score one point for every tail and two points for every head. The person goes on tossing the coin, til his score reaches 100 or exceeds 100. Then the probability that his score attains exactly 100 is
A. `2/3-1/(3.2^(100)`
B. `2/3+1/(3.2^(100))`
C. `2/3-1/(2^(100))`
D. `2/3+1/(2^(100))`

Answer 1

Correct Answer - B
Let `E_(n)` denote the event that the score is `n`
`E_(n)=(E_(n-2)nnH)uu(E_(n-1)nnT)`
`P(E_(n))=1/2[P_(n-2)+P_(n-1)]`
`P_(n)+1/2P_(n-1)=P_(n-1)+1/2P_(n-2)`
`=P_(n-2)+1/2P_(n-3)=P_(2)+1/2P_(1)`
As, `P_(1)=P(T)=1/2impliesP_(2)=P((TnnT)uuH)=3/4`
`impliesP_(n)-2/3=-1/2[P_(n-1)-2/3]=(-1/2)^(2)[P_(n-2)-2/3]`
`=(-1/2)^(n-2)[P_(1)-2/3]=-1/2[1/2-2/3]`
`P_(100)=2/3+1/(3.2^(100))`