For a binomial variate X , if n = 3 and 4P (X=1) = 3P (X=3) , then p (probability of success)

Options

A. A) `1/5`
B. B) `3/5`
C. C) `1/3`
D. D) `2/3`

Answer

To solve the problem, we need to find the probability of success \( p \) for a binomial random variable \( X \) given that \( n = 3 \) and \( 4P(X=1) = 3P(X=3) \).

### Step-by-step Solution:

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

2. **Setting Up the Given Equation**:
We know:
\[ 4P(X=1) = 3P(X=3) \]
We will calculate \( P(X=1) \) and \( P(X=3) \) using the binomial formula.

3. **Calculating \( P(X=1) \)**:
For \( X = 1 \):
\[ P(X=1) = \binom{3}{1} p^1 (1-p)^{3-1} = 3p(1-p)^2 \]

4. **Calculating \( P(X=3) \)**:
For \( X = 3 \):
\[ P(X=3) = \binom{3}{3} p^3 (1-p)^{3-3} = 1 \cdot p^3 \cdot 1 = p^3 \]

5. **Substituting into the Given Equation**:
Now substituting \( P(X=1) \) and \( P(X=3) \) into the equation:
\[ 4(3p(1-p)^2) = 3(p^3) \]
Simplifying this gives:
\[ 12p(1-p)^2 = 3p^3 \]

6. **Dividing Both Sides by \( p \)** (assuming \( p \neq 0 \)):
\[ 12(1-p)^2 = 3p^2 \]

7. **Rearranging the Equation**:
Expanding and rearranging:
\[ 12(1 - 2p + p^2) = 3p^2 \]
\[ 12 - 24p + 12p^2 = 3p^2 \]
\[ 12p^2 - 3p^2 - 24p + 12 = 0 \]
\[ 9p^2 - 24p + 12 = 0 \]

8. **Using the Quadratic Formula**:
We can solve this quadratic equation using the quadratic formula:
\[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 9, b = -24, c = 12 \):
\[ p = \frac{24 \pm \sqrt{(-24)^2 - 4 \cdot 9 \cdot 12}}{2 \cdot 9} \]
\[ p = \frac{24 \pm \sqrt{576 - 432}}{18} \]
\[ p = \frac{24 \pm \sqrt{144}}{18} \]
\[ p = \frac{24 \pm 12}{18} \]
This gives us two possible solutions:
\[ p = \frac{36}{18} = 2 \quad \text{(not valid, since } p \text{ must be between 0 and 1)} \]
\[ p = \frac{12}{18} = \frac{2}{3} \]

9. **Final Answer**:
Therefore, the probability of success \( p \) is:
\[ \boxed{\frac{2}{3}} \]