Bernoulli trials A **Bernoulli trial** is a random experiment with exactly two possible outcomes: success (usually 1) and failure (usually 0). The success probability is p, and the failure probability is 1-p. Binomial experiments A binomial experiment satisfies both conditions: - It consists of n repeated, identically distributed Bernoulli trials, each with success probability p and failure probability 1-p. - The trials are independent, so the outcome of one trial does not affect another. Binomial random variables A binomial random variable X counts the number of successes in n trials with success probability p. X B(n,p), X in 0,1,2,,n Binomial probability distribution function P(X=x)= nxpx(1-p)n-x, x=0,1,2,,n The binomial coefficient nx counts the ways to choose the x successful trials from the n trials.

A **Bernoulli trial** is a random experiment with exactly two possible outcomes: success (usually \(1\)) and failure (usually \(0\)). The success probability is \(p\), and the failure probability is \(1-p\).

A binomial experiment satisfies both conditions:

It consists of \(n\) repeated, identically distributed Bernoulli trials, each with success probability \(p\) and failure probability \(1-p\).

The trials are independent, so the outcome of one trial does not affect another.

A binomial random variable \(X\) counts the number of successes in \(n\) trials with success probability \(p\).

\[X\sim B(n,p),\qquad X\in\{0,1,2,\ldots,n\}\]

Notation and support

Binomial probability distribution function

\[P(X=x)=\binom nxp^x(1-p)^{n-x},\qquad x=0,1,2,\ldots,n\]

BinomPDF

The binomial coefficient \(\binom nx\) counts the ways to choose the \(x\) successful trials from the \(n\) trials.

Binomial Random Variables and Probabilities

Binomial Random Variables and Probabilities