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Bernoulli Trial

TL;DR

Section titled “TL;DR”
  • Each trial has exactly two possible outcomes, commonly labeled “success” and “failure.”
  • The probability of success remains constant across trials, and each trial is independent of the others.
  • Forms the basis for the binomial distribution and calculations of expected value in repeated two-outcome experiments.

Definition

Section titled “Definition”

A Bernoulli trial is a random experiment that has only two possible outcomes, typically referred to as “success” and “failure.” The probability of success and failure is constant for each trial, and the outcome of each trial is independent of the others.

Explanation

Section titled “Explanation”

A Bernoulli trial models a single repetition of a two-outcome process where:

  • Only two outcomes are possible (success/failure).
  • The probability of success does not change from one trial to the next.
  • Outcomes of different trials do not influence each other (independence).

Repeated Bernoulli trials lead to the binomial distribution, which describes the probability of a given number of successes in a fixed number of trials. The expected value is the average outcome over many trials and can be computed directly from the success probability.

Examples

Section titled “Examples”

Coin flip

Section titled “Coin flip”

Flipping a coin yields two possible outcomes, heads or tails, and each outcome has a probability of 0.5. Every time the coin is flipped, the probability of either outcome remains the same, and the outcome of the current flip does not affect the outcome of future flips. This is a Bernoulli trial.

The expected value of the number of heads in a single flip is 0.5.

Die toss

Section titled “Die toss”

The toss of a single die has six possible outcomes, and each outcome has a probability of 1/6. The probability of rolling a certain number does not change, and the outcome of each toss does not affect future tosses. This is also a Bernoulli trial (when considering a specific face as “success”).

The expected value of the number of sixes in a single toss is 1/6.

Multiple coin flips (binomial example)

Section titled “Multiple coin flips (binomial example)”

Flipping a coin 10 times is modeled by the binomial distribution. The binomial distribution describes the probability of flipping a certain number of heads in those 10 flips. For instance:

  • The probability of flipping exactly 5 heads is 0.246.
  • The probability of flipping exactly 6 heads is 0.117.

Real-life examples

Section titled “Real-life examples”
  • Medical trials: success could represent a positive response to a treatment, failure no response or a negative response.
  • Election polls: success could represent a vote for a particular candidate, failure a vote for another candidate or no vote.
  • Gambling games: success could represent a win, failure a loss.

Use cases

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Bernoulli trials are used to model situations with two possible outcomes per trial, such as medical trials, election polls, and gambling games.

Notes or pitfalls

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  • The binomial distribution that arises from repeated Bernoulli trials becomes closer to a normal distribution as the number of trials increases. This property is known as the central limit theorem and is important in many fields, such as statistics and economics.
Section titled “Related terms”
  • Expected value
  • Binomial distribution
  • Central limit theorem