Explanation
$$9p(x = 4) = p(x = 2)$$
$$9{ \times ^6}{C_1}{p^4}{(1 - p)^2}{ = ^6}{C_2}{p^2}{(1 - p)^4}$$
$${{9 \times 6 \times 5} \over 2}{p^2} = {{6 \times 5} \over 2}{\left( {1 - p} \right)^2}$$
$${p^2} = {{{{(1 - p)}^2}} \over 9}$$
$${p \over {1 - p}} = {1 \over 3}$$
$$3p = 1 - p$$
$$4p = 1$$
$$p = {1 \over 4}$$.
$${\textbf{Step -1: Stating the formulas of standard deviation and mean}}{\text{.}}$$
$${\text{We are given }}{(x + y)^{16}}{\text{, therefore, n = 16}}{\text{.}}$$
$${\text{We know that standard variation = }}\sqrt {npq} $$
$$\sqrt {npq} = 2 \Rightarrow npq = 4$$
$${\text{Mean for the distribution = }}np.$$
$${\textbf{Step -2: Finding the value of p}}{\text{.}}$$
$${\text{We know that }}p + q = 1,{\text{so we can write }}q = p - 1.$$
$$\Rightarrow np(1 - p) = 4$$
$$\Rightarrow {\text{16}}p(1 - p) = 4$$
$$\Rightarrow p(1 - p) = \dfrac{1}{4}$$
$$ \Rightarrow p = \dfrac{1}{2}$$
$${\text{Therefore, mean = np = 16}} \times \dfrac{1}{2} = 8.$$
$${\textbf{Hence,}}{\textbf{ The mean of the binomial distribution is 8}}{\textbf{. Thus, option C is correct.}}$$
$${\textbf{Step - 1: Writing given info}}$$
$${\text{Given that the coin is tossed 6 times}}$$
$$\therefore {\text{ Total sample space, n = 6}}$$
$${\text{Probablity of head appearing in one toss, p = }}\dfrac{{\text{1}}}{{\text{2}}}$$
$${\textbf{Step - 2: Calculating variance}}$$
$${\text{Variance = np(1 - p)}}$$
$$ \Rightarrow {\text{ Variance = 6 }} \times {\text{ }}\dfrac{{\text{1}}}{{\text{2}}}{\text{ }} \times {\text{ }}\left( {{\text{1 - }}\dfrac{{\text{1}}}{{\text{2}}}} \right)$$
$$ \Rightarrow {\text{Variance = 6 }} \times {\text{ }}\dfrac{{\text{1}}}{{\text{2}}}{\text{ }} \times {\text{ }}\dfrac{{\text{1}}}{{\text{2}}}$$
$$ \Rightarrow {\text{ Variance = }}\dfrac{{\text{3}}}{{\text{2}}}$$
$$\textbf{Hence option B is correct}$$
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