Question

If \(x\) follows a binomial distribution with parameters \(n = 100\) and \(p = \frac{1}{3}\), then \(P(X = r)\) is maximum when \(r\) equals

• A. 16
• B. 32
• C. 33
• D. None of these

Hint:

Mode of binomial: \(\lfloor (n+1)p \rfloor\) if \((n+1)p\) not integer.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
For binomial, mode = \(\lfloor (n+1)p \rfloor\) or \(\lceil (n+1)p \rceil - 1\).
Step 2: Detailed Explanation:
\((n+1)p = 101 \times \frac{1}{3} = 33.67\)
Since \((n+1)p\) is not an integer, mode = \(\lfloor 33.67 \rfloor = 33\)
Step 3: Final Answer:
\(r = 33\).