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(D) Given that $X \sim B(6, p)$ is a binomial variate with $n=6$ and probability of success $p$. The probability mass function is $P(X=x) = {}^{6}C_{x

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$\frac{1}{4}$

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(D) Given that $X \sim B(6, p)$ is a binomial variate with $n=6$ and probability of success $p$. The probability mass function is $P(X=x) = {}^{6}C_{x} p^{x} q^{6-x}$, where $q = 1-p$. Given $\frac{P(X=4)}{P(X=2)} = \frac{1}{9}$. Substituting the formula: $\frac{{}^{6}C_{4} p^{4} q^{2}}{{}^{6}C_{2} p^{2} q^{4}} = \frac{1}{9}$. Since ${}^{6}C_{4} = {}^{6}C_{2} = 15$, the expression simplifies to: $\frac{p^{2}}{q^{2}} = \frac{1}{9}$. Taking the square root on both sides: $\frac{p}{q} = \frac{1}{3}$ (since $p, q > 0$). Substituting $q = 1-p$: $\frac{p}{1-p} = \frac{1}{3}$. $3p = 1-p \Rightarrow 4p = 1 \Rightarrow p = \frac{1}{4}$.

If $X \sim B(6, p)$ is a binomial variate and $\frac{P(X=4)}{P(X=2)}=\frac{1}{9}$, then $p=$

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