Animated Solution for Mathematics - Probability: The mean and variance of a random variable $X$ having binomial distribution are 4 and 2 respectively, then $P(X=1)$ is

My dear student, welcome to the fascinating world of the Binomial Distribution. Today, we are not just solving a problem; we are peeling back the layers of a mathematical structure that governs everything from coin flips to quality control in manufacturing.

We are given two vital clues: the mean $\mu =4$ and the variance ${\sigma }^2=2$. These are the heartbeat of our distribution.

Imagine you are standing before a mystery box. We know the mean is $np=4$ and the variance is $npq=2$.

The beauty of mathematics lies in its elegance. If we look at the ratio of the variance to the mean, something magical happens:

The $n$ and $p$ terms, which seem so elusive, simply vanish. This leaves us with:

Just like that, the probability of failure is revealed to be $0.5$.

Now that we have $q$, the probability of success $p$ is just a heartbeat away. Because success and failure are complementary, we know that $p+q=1$.

With $q=\frac{1}{2}$, it follows that:

We have discovered that in this specific experiment, success and failure are perfectly balanced, like a fair coin toss.

Now, let us find the total number of trials $n$. Returning to our mean equation, $np=4$, we substitute our value for $p$:

Multiplying both sides by $2$, we find $n=8$. We have successfully mapped the entire landscape of our distribution: $n=8$, $p=0.5$, and $q=0.5$.

The question asks for the probability that $X=1$. We turn to the Binomial Probability Mass Function:

Substituting our values, we get:

Simplifying this, $\left(\genfrac{}{}{0ex}{}{8}{1}\right)$ is simply $8$. The probability terms combine to $(\frac{1}{2}{)}^8$. Thus:

When we divide $8$ by $256$, we arrive at our final, elegant answer:

You see? By breaking the problem down into these logical steps, we didn't just find the answer; we understood the structure of the distribution itself. Keep this clarity with you as you tackle more complex problems. You are doing fantastic.