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

Step 1: Write the two given binomial probabilities.
For $B(5,p)$ with $q=1-p$: \[ P(X=1)=\binom{5}{1}pq^4=5pq^4=0.4096, \] \[ P(X=2)=\binom{5}{2}p^2q^3=10p^2q^3=0.2048. \]
Step 2: Divide to remove constants.
\[ \frac{P(X=2)}{P(X=1)}=\frac{10p^2q^3}{5pq^4}=\frac{2p}{q}. \]
Step 3: Plug in the numbers.
The right side equals $\dfrac{0.2048}{0.4096}=0.5$, so $\dfrac{2p}{q}=0.5$.
Step 4: Solve for $p$.
Then $2p=0.5q=0.5(1-p)$, so $2p=0.5-0.5p$, giving $2.5p=0.5$ and $p=0.2$.
Step 5: Find $q$.
Hence $q=1-0.2=0.8$.
Step 6: Compute the variance.
\[ \text{Var}=npq=5\cdot0.2\cdot0.8=0.64, \] which is option 3.
\[ \boxed{\text{Var}=0.64} \]