Class 12 Statistics Notes · GSEB

Bernoulli Trials

Bernoulli Trials — understand the three essential properties of Bernoulli trials and how they form the foundation of the binomial distribution. GSEB Class 12 Commerce Statistics notes with real-life examples.

Last updated: 22 Sep 2026

Notes

What is a Bernoulli Trial?

Bernoulli Trial

A Bernoulli trial is a random experiment with exactly two possible outcomes: success (with probability p) and failure (with probability q = 1 − p). Each trial is independent and the probability remains constant.

Named after Jacob Bernoulli

The Swiss mathematician Jacob Bernoulli (1655–1705) studied these trials. A single coin toss is the simplest Bernoulli trial — and the foundation of the binomial distribution.

The Three Essential Properties

For a sequence of trials to be Bernoulli trials, all three conditions must hold:

Two Outcomes

Each trial has exactly two results: success or failure. No third option.

Independent Trials

One trial's result does not affect another. The coin doesn't remember the last toss.

Constant Probability

P(success) = p stays the same for every trial. The coin doesn't get tired.

The Notation: p and q

Success — probability p

The outcome we're interested in. It doesn't have to be "good" — in quality control, finding a defective item is also a "success" (because that's what we're checking for).

0 ≤ p ≤ 1

Failure — probability q

The other outcome. Since there are only two possibilities, failure probability is whatever is left.

q = 1 − p

p + q = 1 always

Since success and failure are the only two outcomes, their probabilities must add to 1. If p = 0.7, then q = 0.3. If p = 0.5 (fair coin), then q = 0.5.

Real-Life Bernoulli Trials

Click any example to see how it maps to a Bernoulli trial.

From Bernoulli to Binomial

A single Bernoulli trial is simple. But real-life questions involve repeated trials: "What is the probability of getting 3 heads in 5 tosses?" This is where the binomial distribution comes in.

Bernoulli

1 trial

repeat n times

Binomial

n trials

count successes

X ~ B(n, p)

0, 1, 2, ..., n

Key Takeaways

A Bernoulli trial has exactly two outcomes: success (p) and failure (q = 1 − p).
Three conditions: two outcomes, independent trials, constant probability p.
"Success" is the outcome we track — it doesn't have to be a positive event.
p + q = 1 always. One trial = Bernoulli; n trials = Binomial.
The binomial distribution builds on Bernoulli trials to handle repeated experiments.

Practice

Is drawing a card from a deck a Bernoulli trial? What are success and failure if we're looking for a heart?
A shop has 80% chance of being open on Sunday. Is this a Bernoulli trial? What is p and q?
Why must trials be independent for the binomial distribution to work?