If X follows a binomial distribution with parameters `n=100 and p = 1/3`, then `P(X = r)` is maximum when

Options

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

Answer

To solve the problem, we need to find the value of \( r \) for which \( P(X = r) \) is maximum in a binomial distribution with parameters \( n = 100 \) and \( p = \frac{1}{3} \).

### Step-by-Step Solution:

1. **Identify the parameters of the binomial distribution**:
- The parameters are \( n = 100 \) and \( p = \frac{1}{3} \).

2. **Calculate \( n + 1 \) and \( n + 1 \cdot p \)**:
- First, calculate \( n + 1 \):
\[ n + 1 = 100 + 1 = 101 \]
- Next, calculate \( n + 1 \cdot p \):
\[ n + 1 \cdot p = 101 \cdot \frac{1}{3} = \frac{101}{3} \approx 33.67 \]

3. **Determine if \( n + 1 \cdot p \) is an integer**:
- Since \( \frac{101}{3} \) is not an integer, we proceed to the next step.

4. **Find the unique mode**:
- When \( n + 1 \cdot p \) is not an integer, the unique mode is the integer part of \( n + 1 \cdot p \):
\[ \text{Unique mode} = \lfloor n + 1 \cdot p \rfloor = \lfloor 33.67 \rfloor = 33 \]

5. **Conclusion**:
- Thus, \( P(X = r) \) is maximum when \( r = 33 \).

### Final Answer:
The value of \( r \) for which \( P(X = r) \) is maximum is \( 33 \).