The mean and standard deviation of a binomial distribution are 12 and 2 respectively. What is the number of trials?

Options

A. 2
B. 12
C. 18
D. 24

Explanation

To solve the problem, we need to use the properties of the binomial distribution. The mean (μ) and standard deviation (σ) of a binomial distribution can be expressed in terms of the number of trials (N) and the probability of success (P) as follows:

1. **Mean (μ)**:
\[ \mu = N \cdot P \]
Given that the mean is 12, we have:
\[ N \cdot P = 12 \quad \text{(1)} \]

2. **Standard Deviation (σ)**:
\[ \sigma = \sqrt{N \cdot P \cdot Q} \]
where \( Q = 1 - P \). Given that the standard deviation is 2, we have:
\[ \sqrt{N \cdot P \cdot Q} = 2 \]
Squaring both sides gives:
\[ N \cdot P \cdot Q = 4 \quad \text{(2)} \]

Now we have two equations (1) and (2).

### Step 1: Express Q in terms of P
From equation (1), we can express Q:
\[ Q = 1 - P \]

### Step 2: Substitute Q into equation (2)
Substituting \( Q = 1 - P \) into equation (2):
\[ N \cdot P \cdot (1 - P) = 4 \]

### Step 3: Solve for N
From equation (1), we can express N in terms of P:
\[ N = \frac{12}{P} \]
Substituting this into the modified equation (2):
\[ \frac{12}{P} \cdot P \cdot (1 - P) = 4 \]
This simplifies to:
\[ 12(1 - P) = 4 \]
Solving for P:
\[ 12 - 12P = 4 \\ 12P = 12 - 4 \\ 12P = 8 \\ P = \frac{8}{12} = \frac{2}{3} \]

### Step 4: Find Q
Now, we can find Q:
\[ Q = 1 - P = 1 - \frac{2}{3} = \frac{1}{3} \]

### Step 5: Substitute P back to find N
Now we substitute P back into equation (1) to find N:
\[ N \cdot \frac{2}{3} = 12 \\ N = 12 \cdot \frac{3}{2} = 18 \]

### Conclusion
The number of trials \( N \) is 18.