Question

Suppose $X$ follows a binomial distribution with parameters $n$ and $p$, where $0<p<1$. If $P(X-r)/P(X=n-r)$ is independent of $n$ for every value of $r$, then

• A. $p=1/2$
• B. $p=1/3$
• C. $p=1/4$
• D. $p=1/5$

Answer 1

The correct option is A $p=1/2$

We have
$\frac{P(X=r)}{P(X=n-r)}=\frac{{}^n{C}_r{p}^r(1-p{)}^{n-r}}{{}^n{C}_{n-r}{p}^{n-r}(1-p{)}^r}=\frac{(1-p{)}^{n-2r}}{p^{n-2r}}={(\frac{1}{p}-1)}^{n-2r}$
Note that $\left(1/p\right)-1>0$. Therefore the ratio will be independent of $n$ for each $r$ if $\left(1/p\right)-1=1$, or $p=1/2$.