One hundred identical coins, each with probability 'p' of showing heads are tossed once. If `0 lt p lt 1` and the probability of heads showing on 50 coins is equal to that of heads showing on 51 coins, then the value of p is

Options

A. `(1)/(2)`
B. `(49)/(101)`
C. `(50)/(101)`
D. `(51)/(101)`

Answer

D

Explanation

Let X denotes the number of coins showing heads up. Then, X is a binomial variate with `n=100` and probabilituy of success p.
We have
`P(X=51)=P(X=50)`
`rArr .^100C_(51)p^51q^49=.^100C_(50)p^50q^50`, where `q=1-p`
`rArr (p)/(q)=(.^100C_50)/(.^100C_51)=(51)/(50)rArr (p)/(1-p)=(51)/(50)rArrp=(51)/(101)`