For a binomial variate X , if n = 3 and P(X =1) = 12 P(X =3) , then the value of p is

Options

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

Answer

To solve the problem, we need to find the value of \( p \) given that for a binomial random variable \( X \) with \( n = 3 \), the probability \( P(X = 1) = 12 P(X = 3) \).

### Step-by-Step Solution:

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

2. **Set Up the Equations**:
We know:
\[ P(X = 1) = \binom{3}{1} p^1 (1-p)^{3-1} \]
\[ P(X = 3) = \binom{3}{3} p^3 (1-p)^{3-3} \]
Substituting the values of \( n \):
\[ P(X = 1) = 3p(1-p)^2 \]
\[ P(X = 3) = 1 \cdot p^3 \cdot 1 = p^3 \]

3. **Use the Given Condition**:
According to the problem:
\[ P(X = 1) = 12 P(X = 3) \]
Substituting the expressions we derived:
\[ 3p(1-p)^2 = 12p^3 \]

4. **Simplify the Equation**:
We can divide both sides by \( p \) (assuming \( p \neq 0 \)):
\[ 3(1-p)^2 = 12p^2 \]
Expanding \( (1-p)^2 \):
\[ 3(1 - 2p + p^2) = 12p^2 \]
This simplifies to:
\[ 3 - 6p + 3p^2 = 12p^2 \]
Rearranging gives:
\[ 3 - 6p - 9p^2 = 0 \]
Dividing through by 3:
\[ 1 - 2p - 3p^2 = 0 \]

5. **Rearranging to Standard Form**:
Rearranging gives:
\[ 3p^2 + 2p - 1 = 0 \]

6. **Solving the Quadratic Equation**:
We can use the quadratic formula \( p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 3, b = 2, c = -1 \):
\[ p = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 3 \cdot (-1)}}{2 \cdot 3} \]
\[ p = \frac{-2 \pm \sqrt{4 + 12}}{6} \]
\[ p = \frac{-2 \pm \sqrt{16}}{6} \]
\[ p = \frac{-2 \pm 4}{6} \]
This gives two possible solutions:
\[ p = \frac{2}{6} = \frac{1}{3} \quad \text{or} \quad p = \frac{-6}{6} = -1 \]

7. **Select the Valid Probability**:
Since \( p \) must be between 0 and 1, we discard \( p = -1 \). Thus, the only valid solution is:
\[ p = \frac{1}{3} \]

### Final Answer:
The value of \( p \) is \( \frac{1}{3} \).