# If $X$ follows a binomial distribution with parameters $n$ and $p$. $p \lt 0 \lt 1$ If $P(X=r)/P(X=n-r)$ is independent of $n \:$ and$\: r$ , then value of p is :

$\begin {array} {1 1} (A)\;\large\frac{1}{2} & \quad (B)\;\large\frac{1}{3} \\ (C)\;\large\frac{1}{4} & \quad (D)\;None\: of \: these \end {array}$

$\large\frac{P(x=r)}{P(X=n-r)} = \large\frac{nC_r.p^r.(1-p)^{n-r}}{nC_{n-r}.p^{n-r}.(1-p)^r}$
$= \large\frac{(1-p)^{n-2r}}{p^{n-2r}}$
$= \bigg( \large\frac{1}{p}-1 \bigg)^{n-2r}$ But since $p <1, \: \large\frac{1}{p}$$-1 >0 \therefore The ratio is independent of n and r if \large\frac{1}{p}$$-1=1$
or if $p= \large\frac{1}{2}$
Ans : (A)
edited Mar 26, 2014