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

To find the value of \( r \) at which the probability \( P(X = r) \) is maximum for a binomial distribution, we need to use the formula for the mode of a binomial distribution. For a binomial distribution with parameters \( n \) and \( p \), where \( X \sim B(n, p) \), the mode is generally located at:

• \( \lfloor (n + 1)p \rfloor \) and \( \lfloor (n + 1)p \rfloor - 1 \), where \( \lfloor \cdot \rfloor \) denotes the floor function.

In this specific problem, the parameters are given as \( n = 100 \) and \( p = \frac{1}{3} \).

Let's compute:

1. Calculate \( (n + 1)p \):
   • \( (n + 1)p = (100 + 1) \times \frac{1}{3} = 101 \times \frac{1}{3} = \frac{101}{3} \approx 33.67 \).
2. Apply the floor function:
   • \( \lfloor (n + 1)p \rfloor = \lfloor 33.67 \rfloor = 33 \).
3. Other potential value for maximum probability:
   • \( \lfloor (n + 1)p \rfloor - 1 = 33 - 1 = 32 \).

Thus, the mode of the distribution, or where \( P(X = r) \) is maximum, occurs at \( r = 33 \).

Let's evaluate the options:

• Option 16: Incorrect, as it is not near the calculated mode.
• Option 32: It is a possible candidate but not the highest; 33 yields a higher probability based on integer properties of binomial distribution.
• Option 33: Correct, as this is the calculated maximum mode.
• Option None of these: Incorrect, as 33 is valid.

Thus, the correct answer is 33.