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Question
If X follows a binomial distribution with parameters n = 8 and p = 1/2, then P (|X − 4| ≤ 2) equals
Options
\[\frac{118}{128}\]
\[\frac{119}{128}\]
\[\frac{117}{128}\]
None Of these
Solution
\[\frac{119}{128}\]
\[n = 8, p = \frac{1}{2}\]
\[ \therefore q = 1 - \frac{1}{2} = \frac{1}{2}\]
\[\text{ Hence, the distribution is given by } \]
\[P(X = r) =^{8}{}{C}_r \left( \frac{1}{2} \right)^r \left( \frac{1}{2} \right)^{8 - r} \]
\[P\left( \left| X - 4 \right| \right) \leq 2 \]
\[ = P( - 2 \leq X - 4 \leq 2) \]
\[ = P(2 \leq X \leq 6)\]
\[ = P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6)\]
\[ = \frac{^{8}{}{C}_2 + ^{8}{}{C}_3 + ^{8}{}{C}_4 + ^{8}{}{C}_5 + ^{8}{}{C}_6}{2^8}\]
\[ = \frac{28 + 56 + 70 + 56 + 28}{256}\]
\[ = \frac{238}{256}\]
\[ = \frac{119}{128}\]
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