Binomial DistributionJEE Main

n trials, probability p — interactive Mathematics simulation for IIT-JEE.

Concept

The binomial distribution counts the number of successes in $n$ independent yes/no trials, each succeeding with probability $p$ . It governs anything from coin tosses to defect counts, and its shape shifts from skewed to symmetric as $p$ moves toward $\tfrac12$ .

Key formula

P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}, \quad \mu = np, \quad \sigma^2 = np(1-p)

Derivation

Each specific sequence with $k$ successes and $n-k$ failures has probability $p^k(1-p)^{n-k}$ .

There are $\binom{n}{k}$ such sequences (the ways to choose which trials succeed), so multiply to get the PMF.

Writing $X$ as a sum of $n$ independent Bernoulli variables gives mean $np$ and, by independence, variance $np(1-p)$ .

Scenarios to explore

Binomial Distribution — n trials, probability p — the binomial PMF.

Real-world applications

Quality control — number of defective items in a batch.
Polling and survey sampling.
Genetics — inheritance counts across offspring.

JEE exam tips

Mean is always $np$ ; variance $np(1-p)$ is largest at $p = \tfrac12$ .
For large $n$ the binomial approaches a normal with the same mean and variance.

Common mistakes

Forgetting the binomial coefficient $\binom{n}{k}$ .
Using the binomial when trials are not independent or $p$ is not constant.
Confusing variance $np(1-p)$ with the standard deviation.

Exam traps to avoid

Variance $np(1-p)$ is always less than the mean $np$ (since $1-p<1$ ).
Mode is around $np$ , but use $\lfloor (n+1)p\rfloor$ for the exact most-likely value.