Thank you for reporting, we will resolve it shortly
Q.
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
Probability - Part 2
Solution:
$\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
$(1/p) - 1 = 1 \, \Rightarrow \, p = 1/2$