# Bernoulli Trial: Definition and Examples

Binomial Theorem > Bernoulli Trial

## What is a Bernoulli Trial?

A Bernoulli trial is one of the simplest experiments you can conduct in probability and statistics. It’s an experiment where you can have one of two possible outcomes. For example, “Yes” and “No” or “Heads” and “Tails.” A few more examples:

• Coin tosses: record how many coins land heads up and how many land tails up.
• Births: how many boys are born and how many girls are born each day.
• Rolling Dice: the probability of a roll of two die resulting in a double six.

Coin tossing as a game of probability and chance has been around since Roman times.

Bernoulli trials are usually phrased in terms of success and failure. Success doesn’t mean success in the usual way — it just refers to an outcome you want to keep track of. For example, you might want to find out how many boys are born each day, so you call a boy birth a “success” and a girl birth a “failure.” In the dice rolling example, a double six die roll would be your “success” and everything else rolled would be considered a “failure.”

## What is a Bernoulli Trial: Independence

An important part of every Bernoulli trial is that each action must be independent. That means the probabilities must remain the same throughout the trials; each event must be completely separate and have nothing to do with the previous event.

Winning a scratch off lottery is an independent event. Your odds of winning on one ticket are the same as winning on any other ticket. On the other hand, drawing lotto numbers is a dependent event. Lotto numbers come out of a ball (the numbers aren’t replaced) so the probability of successive numbers being picked depends upon how many balls are left; when there’s a hundred balls, the probability is 1/100 that any number will be picked, but when there are only ten balls left, the probability shoots up to 1/10. While it’s possible to find those probabilities, it isn’t a Bernoulli trial because the events (picking the numbers) are connected to each other.

The Bernouilli process leads to several probability distributions: the binomial distribution, geometric distribution, and negative binomial distribution.

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