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# Suppose a random variable X follows the binomial distribution with parameters n and p,where 0 < p < 1.If P(x=r)/P(x=n-r) is independent of n and r ,then P equals

$\begin{array}{1 1}(A)\;\frac{1}{2}\quad(B)\;\frac{1}{3}\quad(C)\;\frac{1}{5}\quad (D)\;\frac{1}{7}\end{array}$

Toolbox:
• A random variable $X$ following bianomial distribution with parameters $n$ and $p$, its probability distribution is given by
• $\large p(X=r)=c^{n}_{r} p^{r}q^{n-r}$
• When $q=1-p$ and $r=0,1,2,\dots,n$
Given $\huge\frac{p(x=r)}{p(x=n-r)}$ is independent of $n$ and $r$.
$\Rightarrow \huge\frac{c^{n}_{r} p^{r}(1-p)^{n-r}}{c^{n}_{n-r} p^{(n-r)} (1-p)^{r}}$
=$\large p^{2r-n} (1-p)^{n-2r}$
$\Large\left(\frac{1-p}{p}\right)^{n-2r}$
Will be independent of $n$ and $r$ noly if $1-p=p$
$1=2p$
$or \;p=\large\frac{1}{2}$.
$'A'$ option is correct