Question

If X follows a binomial distribution with parameters n = 8 and p = 1/2, then P (|X − 4| ≤ 2) equals
(a) $\frac{118}{128}$

(b) $\frac{119}{128}$

(c) $\frac{117}{128}$

(d) None of these

Answer 1

(b) $\frac{119}{128}$

$$\begin{gathered} n=8,p=\frac{1}{2} \\ \therefore q=1-\frac{1}{2}=\frac{1}{2} \\ \mathrm{Hence},\mathrm{the}\mathrm{distribution}\mathrm{is}\mathrm{given}\mathrm{by} \\ P(X=r)={}^{8}C_r{\left(\frac{1}{2}\right)}^r{\left(\frac{1}{2}\right)}^{8-r} \\ P\left(\left|\mathrm{X}-4\right|\right)\le 2 \\ =\mathit{}P(-2\le X-4\le 2) \\ =P(2\le X\le 6) \\ =P(X=2)+P\mathit{(}X=3)+P(X=4)+P(X=5)+P(X=6) \\ =\frac{{}^{8}C_2+{}^{8}C_3+{}^{8}C_4+{}^{8}C_5+{}^{8}C_6}{2^8} \\ =\frac{28+56+70+56+28}{256} \\ =\frac{238}{256} \\ =\frac{119}{128} \end{gathered}$$