Question

The probability of success is p in one trial and the experiment be repeated n times. If the ratio of the probability of exactly r success to the probability of exactly r failures is independent of n and r, then p =

• A. $\frac{1}{2}$
• B. exists no such value of p
• C. $\frac{1}{3}$
• D. None of these

Answer 1

The correct option is A $\frac{1}{2}$

Probability of r successes in n trials is equal to ${}^n{C}_r{p}^r{q}^{n-r}$, where p is the probability of success in one trial and q is the probability of failure in one trial.
$\frac{{}^n{C}_r{p}^r(1-p{)}^{n-r}}{{}^n{C}_r{p}^{n-r}(1-p{)}^r}={\left(\frac{p}{1-p}\right)}^{2r-n}$
is independent of n and r. For this to be indepent of n and r, we should have
$\frac{p}{1-p}=1\Rightarrow p=\frac{1}{2}$