Question

If the mean and variance of a binomial distribution are 4 and 2 respectively, then probability of getting 2 successes is

• A. $\frac{28}{256}$
• B. $\frac{37}{256}$
• C. $\frac{128}{256}$
• D. $\frac{219}{256}$

Hint:

Whenever the variance is exactly half of the mean ($npq = \frac{1}{2}np$), the distribution is perfectly symmetric, meaning $p = q = \frac{1}{2}$. This simplifies the calculation because the exponential term is always constant: $p^k q^{n-k} = \left(\frac{1}{2}\right)^n$. The final numerator is simply the combination value $\binom{n}{k}$!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given the mean (4) and variance (2) of a standard binomial distribution parameter group. We need to calculate the exact probability of obtaining exactly 2 successes, denoted as $P(X=2)$.

Step 2: Key Formula or Approach:
For a binomial distribution $X \sim B(n,p)$ with $n$ trials and success probability $p$, the statistical formulas are: $$\text{Mean} = np$$ $$\text{Variance} = npq$$ where $q = 1 - p$ is the probability of failure. The probability of getting exactly $k$ successes is given by the formula: $$P(X = k) = \binom{n}{k} p^k q^{n-k}$$

Step 3: Detailed Explanation:
Given the mean and variance equations: $$np = 4 \quad \text{--- (Equation 1)}$$ $$npq = 2 \quad \text{--- (Equation 2)}$$ To isolate $q$, divide Equation 2 by Equation 1: $$\frac{npq}{np} = \frac{2}{4} \implies q = \frac{1}{2}$$ Since $p = 1 - q$: $$p = 1 - \frac{1}{2} = \frac{1}{2}$$ Now, substitute $p = \frac{1}{2}$ back into Equation 1 to find the total number of trials $n$: $$n\left(\frac{1}{2}\right) = 4 \implies n = 8$$ Our binomial distribution is fully defined by $n = 8$ and $p = q = \frac{1}{2}$. We need to calculate the probability of getting exactly $k = 2$ successes: $$P(X = 2) = \binom{8}{2} \left(\frac{1}{2}\right)^2 \left(\frac{1}{2}\right)^{8-2}$$ $$P(X = 2) = \binom{8}{2} \left(\frac{1}{2}\right)^8$$ Calculate the combination value $\binom{8}{2}$: $$\binom{8}{2} = \frac{8 \times 7}{2 \times 1} = 28$$ Calculate the exponential denominator term $2^8$: $$2^8 = 256$$ Combine these values to get the final fraction: $$P(X = 2) = \frac{28}{256}$$

Step 4: Final Answer:
The probability of getting 2 successes is $\frac{28}{256}$, which corresponds to option (A).