If for a binomial distribution with $n = 5, 4 P (X = 1) = P (X = 2),$ the probability of success is

\frac{1}{3}

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\frac{2}{3}

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\frac{1}{4}

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\frac{1}{8}

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Solution

The correct option is C $\frac{2}{3}$
$n = 5,$ $4 P (X = 1) = P (X = 2)$
$P (X = k)$ $=^{n} C_{k} p^{k} {(1 - p)}^{n - k}$
$=> 4 P (X = 1) = P (X = 2)$
$=> 4$ $^{5} C_{1} p^{1} {(1 - p)}^{5 - 1} =^{5} C_{2} p^{2} {(1 - p)}^{5 - 2}$
$=> 20$ $p {(1 - p)}^{4} = 10 p^{2} {(1 - p)}^{3}$
$=>$ $20 (1 - p) = 10 p$
$=> p =$ $\frac{2}{3}$

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