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 and r, then

• A. $p=\frac{1}{2}$
• B. $p=\frac{1}{3}$
• C. $p=\frac{1}{4}$
• D. none of these

Answer 1

Answer: (A)

$\frac{P\left(X=r\right)}{P\left(X=n-r\right)}=\frac{{}^n{C}_r{p}^r{\left(1-p\right)}^{n-r}}{{}^n{C}_{n-r}{p}^{n-r}{\left(1-p\right)}^r}$
$=\frac{{\left(1-p\right)}^{n-2r}}{p^{n-2r}}$
$={\left(\frac{1-p}{p}\right)}^{n-2r}={\left(\frac{1}{p}-1\right)}^{n-2r}$
and $\left(\frac{1}{p}\right)-1>0$
$\therefore$ ratio will be independent of n and r if
$(1/p)-1=1\Rightarrow p=1/2$