Question

If \( X \sim B(4, p) \) and \( 2 P(X = 3) = 3 P(X = 2) \), then the value of \( p \) is:

• A. \( \frac{9}{13} \)
• B. \( \frac{4}{13} \)
• C. \( \frac{1}{13} \)
• D. \( \frac{12}{13} \)

Hint:

For binomial probability problems, always start by writing down the general formula for \( P(X = k) \) and use the given conditions to set up an equation.

Answer 1

The Correct Option is A

Step 1: Binomial distribution formula.
The probability mass function for a binomial distribution is given by: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] For \( X \sim B(4, p) \), the probabilities are: \[ P(X = 3) = \binom{4}{3} p^3 (1-p)^1 = 4p^3(1-p) \] \[ P(X = 2) = \binom{4}{2} p^2 (1-p)^2 = 6p^2(1-p)^2 \] Step 2: Apply the given condition.
We are given that \( 2 P(X = 3) = 3 P(X = 2) \). Substituting the probabilities: \[ 2 \times 4p^3(1-p) = 3 \times 6p^2(1-p)^2 \] Simplify: \[ 8p^3(1-p) = 18p^2(1-p)^2 \] Cancel \( (1-p) \) from both sides (assuming \( p \neq 1 \)): \[ 8p^3 = 18p^2(1-p) \] Divide by \( p^2 \) (assuming \( p \neq 0 \)): \[ 8p = 18(1 - p) \] Simplifying: \[ 8p = 18 - 18p \] \[ 26p = 18 \] \[ p = \frac{9}{13} \] Step 3: Conclusion.
Thus, the value of \( p \) is \( \boxed{\frac{9}{13}} \).