Question

In a binomial distribution, the mean is 4 and the variance is 3. Then the number of trials $n$ is:

• A. 8
• B. 12
• C. 16
• D. 20

Hint:

In a binomial distribution, the variance is always less than the mean. The ratio $\text{Variance}/\text{Mean}$ gives $q$ directly, so $p = 1 - q$, and then $n = \text{Mean}/p$.

Answer 1

The Correct Option is C

Step 1: Binomial mean and variance.
For a binomial distribution, mean $= np$ and variance $= npq$, where $q = 1 - p$.

Step 2: Write the givens.
$np = 4$ and $npq = 3$.

Step 3: Divide to find $q$.
\[ \frac{npq}{np} = \frac{3}{4} \implies q = \frac34 \]

Step 4: Find $p$.
Since $p = 1 - q$: \[ p = 1 - \frac34 = \frac14 \]

Step 5: Use the mean again.
\[ n \times \frac14 = 4 \]

Step 6: Solve for $n$.
\[ n = 16 \] \[ \boxed{ n = 16 } \]