A binomial random variable X, 5P(X = 3) = 2P(X = 2), when n = 5. Find the value of parameter p.

Answer

To solve the problem, we need to find the value of the parameter \( p \) in a binomial distribution where \( n = 5 \) and the relationship between the probabilities of \( X = 3 \) and \( X = 2 \) is given by \( 5P(X = 3) = 2P(X = 2) \).

### Step-by-Step Solution:

1. **Understand the Binomial Probability Formula**:
The probability of getting exactly \( k \) successes in \( n \) trials is given by:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]
where \( \binom{n}{k} \) is the binomial coefficient.

2. **Set Up the Probabilities**:
For \( n = 5 \):
- Calculate \( P(X = 3) \):
\[ P(X = 3) = \binom{5}{3} p^3 (1-p)^{2} \]
- Calculate \( P(X = 2) \):
\[ P(X = 2) = \binom{5}{2} p^2 (1-p)^{3} \]

3. **Substitute the Binomial Coefficients**:
- The binomial coefficients are:
\[ \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \]
\[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

Thus, we have:
\[ P(X = 3) = 10 p^3 (1-p)^2 \]
\[ P(X = 2) = 10 p^2 (1-p)^3 \]

4. **Set Up the Given Equation**:
According to the problem:
\[ 5P(X = 3) = 2P(X = 2) \]
Substituting the probabilities:
\[ 5(10 p^3 (1-p)^2) = 2(10 p^2 (1-p)^3 \]
Simplifying gives:
\[ 50 p^3 (1-p)^2 = 20 p^2 (1-p)^3 \]

5. **Cancel Common Terms**:
Assuming \( p \neq 0 \) and \( 1-p \neq 0 \), we can divide both sides by \( 10p^2(1-p)^2 \):
\[ 5p = 2(1-p) \]

6. **Rearranging the Equation**:
Expanding and rearranging gives:
\[ 5p = 2 - 2p \]
\[ 5p + 2p = 2 \]
\[ 7p = 2 \]

7. **Solve for \( p \)**:
\[ p = \frac{2}{7} \]

### Final Answer:
The value of the parameter \( p \) is \( \frac{2}{7} \).