Animated Solution for Mathematics - Probability: A dice is tossed 5 times. Getting an odd number is considered a success. Then the variance of distribution of success is

Imagine you are holding a standard six-sided die in your hand. You are about to embark on a journey of five independent tosses.

Each toss is a moment of pure chance, yet within this randomness lies a beautiful, predictable structure. This is the essence of the Binomial Distribution, a cornerstone of probability theory that allows us to quantify uncertainty.

We are tossing a die $n=5$ times. Each toss is an independent trial, meaning the outcome of one toss does not influence the next.

This independence is crucial. It allows us to treat the entire experiment as a sequence of Bernoulli trials, where each trial has only two possible outcomes: success or failure.

The problem defines success as getting an odd number. On a standard die, the possible outcomes are $\{1,2,3,4,5,6\}$.

The odd numbers are $\{1,3,5\}$. Thus, there are 3 favorable outcomes. The probability of success $p$ is the ratio of favorable outcomes to total outcomes:

Conversely, the probability of failure $q$ is the complement of success:

Now that we have our parameters—$n=5$, $p=\frac{1}{2}$, and $q=\frac{1}{2}$—we can use the power of the Binomial Distribution. This distribution is designed to model the number of successes in a fixed number of independent trials.

One of its most elegant properties is the formula for variance:

This formula tells us how much the number of successes is expected to deviate from the mean. It is a measure of the spread or uncertainty in our experiment.

With the formula $Var(X)=npq$ in hand, the final step is a simple substitution. We have $n=5$, $p=\frac{1}{2}$, and $q=\frac{1}{2}$.

Substituting these into the formula, we get:

Performing the multiplication, we multiply the numerators: $5\times 1\times 1=5$. Then, we multiply the denominators: $2\times 2=4$.

The result is:

And there you have it! The variance of the distribution of success is $\frac{5}{4}$.

This journey through the problem shows that even in a series of random dice tosses, we can find order and precision using the tools of probability. Keep practicing, and you will find that these concepts become second nature.