Question:
The mean and variance of a binomial distribution are 8 and 4 respectively. What is \( P(X=1) \)?
The mean and variance of a binomial distribution are 8 and 4 respectively. What is \( P(X=1) \)?
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Always derive \( p, q, n \) first before applying binomial probability formula.
Updated On: May 1, 2026
- \( \frac{1}{2^8} \)
- \( \frac{1}{2^{12}} \)
- \( \frac{1}{2^6} \)
- \( \frac{1}{2^4} \)
- \( \frac{1}{2^5} \)
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The Correct Option is A
Solution and Explanation
Concept:
For binomial distribution:
\[
\text{Mean} = np,\quad \text{Variance} = npq
\]
Step 1: Use given values.
\[ np = 8,\quad npq = 4 \]
Step 2: Divide equations.
\[ q = \frac{4}{8} = \frac{1}{2} \Rightarrow p = \frac{1}{2} \]
Step 3: Find \( n \).
\[ np = 8 \Rightarrow n = 16 \]
Step 4: Use binomial formula.
\[ P(X=1) = \binom{16}{1} \left(\frac{1}{2}\right)^1 \left(\frac{1}{2}\right)^{15} \]
Step 5: Simplify.
\[ = 16 \cdot \frac{1}{2^{16}} = \frac{1}{2^8} \]
Step 1: Use given values.
\[ np = 8,\quad npq = 4 \]
Step 2: Divide equations.
\[ q = \frac{4}{8} = \frac{1}{2} \Rightarrow p = \frac{1}{2} \]
Step 3: Find \( n \).
\[ np = 8 \Rightarrow n = 16 \]
Step 4: Use binomial formula.
\[ P(X=1) = \binom{16}{1} \left(\frac{1}{2}\right)^1 \left(\frac{1}{2}\right)^{15} \]
Step 5: Simplify.
\[ = 16 \cdot \frac{1}{2^{16}} = \frac{1}{2^8} \]
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