Question

In a binomial distribution with 5 trials, the probabilities of 1 success and 2 successes are 0.4096 and 0.2048 respectively. Find the variance of the distribution.

• A. 0.80
• B. 0.75
• C. 0.64
• D. 0.72

Hint:

The ratio method $\frac{P(X=k+1)}{P(X=k)}$ is highly effective for binomial problems because the binomial coefficients and powers of $p, q$ simplify drastically.

Answer 1

The Correct Option is C

Concept: For a binomial distribution $B(n, p)$, $P(X=k) = \binom{n}{k} p^k q^{n-k}$, where $q=1-p$. The variance is given by $Var(X) = npq$.

Step 1: Set up the binomial probability equations.
$P(X=1) = \binom{5}{1} p^1 q^4 = 5pq^4 = 0.4096$
$P(X=2) = \binom{5}{2} p^2 q^3 = 10p^2q^3 = 0.2048$

Step 2: Take the ratio to eliminate one variable.
\[ \frac{P(X=2)}{P(X=1)} = \frac{10p^2q^3}{5pq^4} = \frac{2p}{q} \] \[ \frac{2p}{q} = \frac{0.2048}{0.4096} = 0.5 \] \[ 2p = 0.5q \implies 2p = 0.5(1-p) \] \[ 2p = 0.5 - 0.5p \implies 2.5p = 0.5 \implies p = \frac{0.5}{2.5} = 0.2 \]

Step 3: Calculate variance \(Var(X) = npq\).
$n = 5, p = 0.2, q = 1 - 0.2 = 0.8$.
\[ Var(X) = 5 \cdot 0.2 \cdot 0.8 = 1.0 \cdot 0.8 = 0.64 \] center minipage0.3

Variance = 0.64 minipage center