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Binomial Probability theorem

The probabili...

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 =

\frac{1}{2}

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exists no such value of p

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\frac{1}{3}

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None of these

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Solution

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.
$\frac{^{n} C_{r} p^{r} (1 - p)^{n - r}}{^{n} C_{r} p^{n - r} (1 - p)^{r}} = {(\frac{p}{1 - p})}^{2 r - n}$
is independent of n and r. For this to be indepent of n and r, we should have
$\frac{p}{1 - p} = 1 \Rightarrow p = \frac{1}{2}$

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