Let `X ~ B( n , p )` , If `n = 10 `and `E(X) = 5` , Using the following activity find p and Var (X).

Answer

To solve the problem, we need to find the probability \( p \) and the variance \( Var(X) \) for a binomial distribution \( X \sim B(n, p) \) where \( n = 10 \) and \( E(X) = 5 \).

### Step-by-Step Solution:

1. **Understand the Given Information:**
- We have a binomial distribution \( X \) with parameters \( n \) (number of trials) and \( p \) (probability of success).
- Given \( n = 10 \) and \( E(X) = 5 \).

2. **Use the Formula for Expected Value:**
- The expected value \( E(X) \) for a binomial distribution is given by:
\[ E(X) = n \cdot p \]
- Substituting the known values:
\[ 5 = 10 \cdot p \]

3. **Solve for \( p \):**
- Rearranging the equation to find \( p \):
\[ p = \frac{5}{10} = \frac{1}{2} \]

4. **Find the Value of \( q \):**
- The value of \( q \) is defined as:
\[ q = 1 - p \]
- Substituting the value of \( p \):
\[ q = 1 - \frac{1}{2} = \frac{1}{2} \]

5. **Use the Formula for Variance:**
- The variance \( Var(X) \) for a binomial distribution is given by:
\[ Var(X) = n \cdot p \cdot q \]
- Substituting the known values:
\[ Var(X) = 10 \cdot \frac{1}{2} \cdot \frac{1}{2} \]

6. **Calculate the Variance:**
- Performing the multiplication:
\[ Var(X) = 10 \cdot \frac{1}{4} = \frac{10}{4} = 2.5 \]

### Final Answers:
- The probability \( p \) is \( \frac{1}{2} \).
- The variance \( Var(X) \) is \( 2.5 \).