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Bernoulli Trials: Definition, Examples

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What are Bernoulli Trials?

bernoulli trials

A coin toss is a Bernoulli trial with a probability of heads = 0.5.

A Bernoulli trial is an experiment with two possible outcomes: Success or Failure. “Success” in one of these trials means that you’re getting the result you’re measuring. For example:

  • If you flip a coin 100 times to see how many heads you get, then the Success is getting heads and a Failure is getting tails.
  • You might want to find out how many girls are born each day, so a girl birth is a Success and a boy birth is a Failure.
  • You want to find the probability of rolling a double six in a dice game. A double six dice roll = Success and everything else = Failure.

Note that “Success” doesn’t have the traditional meaning of triumph or prosperity. In the context of Bernoulli trials, it’s merely a way of counting the result you’re interested in. For example, you might want to know how many students get the last question on a test wrong. As you’re measuring the number of incorrect answers, the “Success” is answered incorrectly and a “Failure” is answered correctly. Of course, this might get confusing so there’s nothing stopping you tweaking your hypothesis so that you’re measuring the number of correct answers instead of incorrect ones.

Probability Distribution for Bernoulli Trials

Bernoulli trials are a special case of i.i.d. trials; Trials are i.i.d. if all the random variables in the trials have the same probability distribution.

The probability distribution for a Bernoulli trial is given by the binomial probability distribution:


  • ! is a factorial,
  • x is the number of successes,
  • n is the number of trials.

Assumptions for Bernoulli Trials

The three assumptions for Bernoulli trials are:

  1. Each trial has two possible outcomes: Success or Failure. We are interested in the number of Successes X (X = 0, 1, 2, 3,…).
  2. The probability of Success (and of Failure) is constant for each trial; a “Success” is denoted by the letter p and “Failure” is q = 1 − p.
  3. Each trial is independent; The outcome of previous trials has no influence on any subsequent trials.


Governors State University. General PPT.

Stephanie Glen. "Bernoulli Trials: Definition, Examples" From Elementary Statistics for the rest of us!

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