Binomial Distribution

• Definition: The distribution of the total number of successes in $n$ independent Bernoulli trials, each with success probability $p$.
• Mean: $E[X]=np$
• Variance: $Var(X)=np(1-p)$
• Use case: Models repeated independent yes/no events (How many heads in 10 coin flips?).

Represented as $\mathrm{Binomial}(p)$

Mathematically as $P(x)=\left(\genfrac{}{}{0ex}{}{n}{x}\right)p^x{q}^{n-x}$ where

• $n$: the number of trials
• $x$: the number of successes desired
• $p$: probability of getting a success in one trial
• $q=1-p$: probability of getting a failure in one trial

Visually as