Question

If \(X\) follows a binomial distribution with parameters \(n = 8\) and \(p = \frac{1}{2}\), then \(P(|x - 4| \le 2)\) is equal to

• A. \(\frac{119}{128}\)
• B. \(\frac{116}{128}\)
• C. \(\frac{29}{128}\)
• D. None of these

Hint:

For binomial distribution with \(p=1/2\), \(P(X=r) = \binom{n}{r}/2^n\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
For binomial distribution, \(P(X = r) = \binom{n}{r} p^r q^{n-r}\). Here \(n=8\), \(p=q=1/2\).

Step 2: Detailed Explanation:
\(|x-4| \le 2\) means \(x = 2, 3, 4, 5, 6\). \(P(X = r) = \binom{8}{r} \left(\frac{1}{2}\right)^8 = \frac{\binom{8}{r}}{256}\). \[ P(2) = \frac{28}{256},\ P(3) = \frac{56}{256},\ P(4) = \frac{70}{256},\ P(5) = \frac{56}{256},\ P(6) = \frac{28}{256} \] Sum = \(\frac{28+56+70+56+28}{256} = \frac{238}{256} = \frac{119}{128}\).

Step 3: Final Answer:
\(\frac{119}{128}\), which corresponds to option (A).