Correct Answer - Option 4 : 137
Given, the mean, μ = 8
and the variance is σ2 = 4
Let number of trials be n
Let probability of success be p
Let probability of failure be q
We know that, for Binomial distribution,
Mean μ = np,
⇒ np = 8
Variance, σ2 = npq
⇒ npq = 4
Now, the value of ‘q’ is:
⇒ (8)q = 4
\(\Rightarrow q = \frac{4}{8}\)
\(\therefore q = \frac{1}{2}\)
∵ [p + q = 1]
Now, the value of ‘p’ is:
\(\Rightarrow {\rm{p}} + \frac{1}{2} = 1\)
\(\Rightarrow p = 1 - \frac{1}{2}\)
\(\therefore p = \frac{1}{2}\)
Now, the value of ‘n’ is:
\(\Rightarrow n\left( {\frac{1}{2}} \right) = 8\)
∴ n = 16
Given, \(P\left( {X \le 2} \right) = \frac{k}{{{2^{16}}}}\)
\(\Rightarrow P\left( {X = 0} \right) + P\left( {X = 1} \right) + P\left( {X = 2} \right) = \frac{k}{{{2^{16}}}}\)
\(\Rightarrow P\left( {X \le 2} \right) = \frac{{{}_{\rm{\;}}^{16}{C_0} + {}_{\rm{\;}}^{16}{C_1} + {}_{\rm{\;}}^{16}{C_2}}}{{{2^{16}}}} = \frac{k}{{{2^{16}}}}\)
∵ \({}_{\rm{\;}}^{16}{C_0} = \frac{{16!}}{{0!\left( {16 - 0} \right)!}} = \frac{{16!}}{{16!}}\)
\({}_{\rm{\;}}^{16}{C_1} = \frac{{16!}}{{1!\left( {16 - 1} \right)!}} = \frac{{16!}}{{15!}} = \frac{{16 \times 15!}}{{15!}} = 16\)
\({}_{\rm{}}^{16}{C_2} = \frac{{16!}}{{2!\left( {16 - 2} \right)!}} = \frac{{16!}}{{2! \times 14!}}\)
\(= \frac{{16 \times 15 \times 14!}}{{2 \times 14!}} = \frac{{16 \times 15}}{2} = 8 \times 15 = 120\)
\(\Rightarrow P\left( {X \le 2} \right) = \frac{{1 + 16 + 120}}{{{2^{16}}}} = \frac{k}{{{2^{16}}}}\)
\(\Rightarrow P\left( {X \le 2} \right) = \frac{{137}}{{{2^{16}}}} = \frac{k}{{{2^{16}}}}\)
∴ k = 137