The mean and the variance of a binomial distribution are 4 and 2 respectively. Then, the probability of 2 successes is

Options

A. `(28)/(256)`
B. `(219)/(256)`
C. `(128)/(256)`
D. `(37)/(256)`

Explanation

To solve the problem, we need to find the probability of getting exactly 2 successes in a binomial distribution where the mean (μ) is 4 and the variance (σ²) is 2.

### Step-by-Step Solution:

1. **Understand the parameters of the binomial distribution**:
- The mean (μ) of a binomial distribution is given by the formula:
\[ \mu = n \cdot p \]
- The variance (σ²) is given by:
\[ \sigma^2 = n \cdot p \cdot q \]
where \( q = 1 - p \).

2. **Set up the equations**:
- From the mean, we have:
\[ n \cdot p = 4 \quad \text{(1)} \]
- From the variance, we have:
\[ n \cdot p \cdot q = 2 \quad \text{(2)} \]
- Since \( q = 1 - p \), we can substitute \( q \) in equation (2):
\[ n \cdot p \cdot (1 - p) = 2 \quad \text{(3)} \]

3. **Substituting equation (1) into equation (3)**:
- From equation (1), we can express \( n \) in terms of \( p \):
\[ n = \frac{4}{p} \]
- Substitute \( n \) into equation (3):
\[ \frac{4}{p} \cdot p \cdot (1 - p) = 2 \]
- Simplifying gives:
\[ 4(1 - p) = 2 \]
- Solving for \( p \):
\[ 4 - 4p = 2 \implies 4p = 2 \implies p = \frac{1}{2} \]

4. **Finding \( n \)**:
- Substitute \( p \) back into equation (1):
\[ n \cdot \frac{1}{2} = 4 \implies n = 8 \]

5. **Calculate the probability of getting exactly 2 successes**:
- The probability of getting exactly \( r \) successes in a binomial distribution is given by:
\[ P(X = r) = \binom{n}{r} p^r q^{n-r} \]
- Here, \( n = 8 \), \( r = 2 \), \( p = \frac{1}{2} \), and \( q = \frac{1}{2} \):
\[ P(X = 2) = \binom{8}{2} \left(\frac{1}{2}\right)^2 \left(\frac{1}{2}\right)^{8-2} \]
- This simplifies to:
\[ P(X = 2) = \binom{8}{2} \left(\frac{1}{2}\right)^8 \]

6. **Calculate \( \binom{8}{2} \)**:
- \(\binom{8}{2} = \frac{8 \times 7}{2 \times 1} = 28\)

7. **Final calculation**:
- Therefore:
\[ P(X = 2) = 28 \cdot \left(\frac{1}{2}\right)^8 = 28 \cdot \frac{1}{256} = \frac{28}{256} = \frac{7}{64} \]

### Final Answer:
The probability of getting exactly 2 successes is \( \frac{7}{64} \).