Let the random variable X follow B(6.p). If 16P(X=4)=P(X=2), then what is the value of p?

Options

A. `(1)/(3)`
B. `(1)/(4)`
C. `(1)/(5)`
D. `(1)/(6)`

Answer

To solve the problem, we need to find the value of \( p \) given that the random variable \( X \) follows a binomial distribution \( B(6, p) \) and that \( 16P(X=4) = P(X=2) \).

### Step-by-Step Solution:

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

2. **Set Up the Equations**:
For \( n = 6 \):
- \( P(X = 4) = \binom{6}{4} p^4 (1-p)^{2} \)
- \( P(X = 2) = \binom{6}{2} p^2 (1-p)^{4} \)

Given that \( 16P(X=4) = P(X=2) \), we can write:
\[ 16 \cdot \binom{6}{4} p^4 (1-p)^2 = \binom{6}{2} p^2 (1-p)^4 \]

3. **Calculate the Binomial Coefficients**:
- \( \binom{6}{4} = \binom{6}{2} = 15 \)

Thus, the equation simplifies to:
\[ 16 \cdot 15 p^4 (1-p)^2 = 15 p^2 (1-p)^4 \]

4. **Cancel Out Common Terms**:
We can divide both sides by \( 15 \) (assuming \( p^2(1-p)^2 \neq 0 \)):
\[ 16 p^4 (1-p)^2 = p^2 (1-p)^4 \]

5. **Rearranging the Equation**:
Rearranging gives:
\[ 16 p^4 = p^2 (1-p)^2 \]

6. **Divide by \( p^2 \)** (assuming \( p \neq 0 \)):
\[ 16 p^2 = (1-p)^2 \]

7. **Expand and Rearrange**:
Expanding the right side:
\[ 16 p^2 = 1 - 2p + p^2 \]
Rearranging gives:
\[ 15 p^2 + 2p - 1 = 0 \]

8. **Use the Quadratic Formula**:
The quadratic formula \( p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) applies here with \( a = 15, b = 2, c = -1 \):
\[ p = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 15 \cdot (-1)}}{2 \cdot 15} \]
\[ p = \frac{-2 \pm \sqrt{4 + 60}}{30} \]
\[ p = \frac{-2 \pm \sqrt{64}}{30} \]
\[ p = \frac{-2 \pm 8}{30} \]

9. **Calculate the Roots**:
- First root: \( p = \frac{6}{30} = \frac{1}{5} \)
- Second root: \( p = \frac{-10}{30} = -\frac{1}{3} \) (not valid since probability cannot be negative)

10. **Conclusion**:
The valid solution is:
\[ p = \frac{1}{5} \]

### Final Answer:
The value of \( p \) is \( \frac{1}{5} \).