In a binomial distribution the mean is 20, standard deviation is `sqrt15` and p =

Options

A. `(3)/(4)`
B. `(1)/(4)`
C. `(1)/(2)`
D. `(1)/(3)`

Answer

To solve the problem, we need to find the value of \( p \) in a binomial distribution given the mean and standard deviation.

### Step-by-Step Solution:

1. **Understand the Mean and Standard Deviation in a Binomial Distribution:**
- The mean \( \mu \) of a binomial distribution is given by:
\[ \mu = n \cdot p \]
- The standard deviation \( \sigma \) is given by:
\[ \sigma = \sqrt{n \cdot p \cdot (1 - p)} \]

2. **Given Values:**
- From the problem, we have:
\[ \mu = 20 \quad \text{and} \quad \sigma = \sqrt{15} \]

3. **Set Up the Equations:**
- From the mean:
\[ n \cdot p = 20 \quad \text{(1)} \]
- From the standard deviation:
\[ \sigma^2 = n \cdot p \cdot (1 - p) \implies 15 = n \cdot p \cdot (1 - p) \quad \text{(2)} \]

4. **Substitute Equation (1) into Equation (2):**
- From equation (1), we can express \( n \) in terms of \( p \):
\[ n = \frac{20}{p} \]
- Substitute \( n \) into equation (2):
\[ 15 = \left(\frac{20}{p}\right) \cdot p \cdot (1 - p) \]
- Simplifying this gives:
\[ 15 = 20(1 - p) \]

5. **Solve for \( p \):**
- Rearranging the equation:
\[ 15 = 20 - 20p \]
\[ 20p = 20 - 15 \]
\[ 20p = 5 \]
\[ p = \frac{5}{20} = \frac{1}{4} \]

6. **Final Answer:**
- The value of \( p \) is:
\[ p = \frac{1}{4} \]