Bernoulli Distribution: Definition, Formula, and Real-World Examples

Key takeaways

• The Bernoulli distribution is a discrete probability distribution with outcomes that take on specific values, making it suitable for random events with binary outcomes.

• To represent the random variable X, you can use the Bernoulli distribution formula: P(X = x) = p^x (1 - p)^(1 = x), for x = 0 or x = 1.

• The distinction between the Bernoulli and binomial distributions is that the Bernoulli distribution describes a single trial with two possible outcomes, while a binomial distribution represents the number of successes across multiple repeated Bernoulli trials.

You can use the Bernoulli distribution when you have two clear outcomes, and each trial is independent of the others with a consistent probability of success. Explore the Bernoulli distribution, including formulas and examples, to better understand how to read and use related equations and how it applies across industries.

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What is the Bernoulli distribution?

The Bernoulli distribution is a type of discrete probability distribution used for random experiments with exactly two possible outcomes. You can think of things like coin tosses (heads/tails), passing an exam (pass/fail), or answering a question correctly (correct/incorrect).

Typically, one outcome is a success and one a failure, represented by “1” and “0.” The p-value related to this distribution represents the probability that the success outcome happens.

The probability distribution is considered “discrete,” meaning that the outcomes take on specific, separate values rather than “continuous,” which means that they fall on a continuous spectrum. For example, if you flip a coin, you either get heads or tails. You can’t get half of a tail and half of a head, or one-third of a tail and two-thirds of a tail [1].

Bernoulli vs. binomial distribution: What’s the difference?

The Bernoulli distribution describes a single trial outcome with two possible results, whereas a binomial distribution represents the number of successes across many repeated Bernoulli trials. For example, one coin flip follows the Bernoulli distribution, while the number of heads in 10 coin flips follows a binomial distribution [2]. 

Read more: Is Data Analytics Hard? Tips for Rising to the Challenge

What is a Bernoulli trial?

A Bernoulli trial is a single experiment with exactly two outcomes, a fixed probability of success (p), and results that are independent of all other trials.

One coin flip is a Bernoulli trial. The two outcomes are heads and tails, the “p” of success is 0.5, and one result does not influence the outcome of other coin flips [1].

What is the Bernoulli distribution formula?

The probability mass function (PMF) represents the probability of each possible value the random variable can take (a coin could be your random variable). This PMF captures the random variable's behavior. For the Bernoulli distribution, you represent the random variable X by the formula [3]:

P(X = x) = p^x (1 - p)^(1 = x), for x = 0 or x = 1

In this equation:

• p is the probability of success

• x is the outcome of interest

While it may look complicated, when you put in the “1” and “0” values, you can simplify the versions: P(X = 1) = p

P(X = 0) = 1 - p

This tells you that the probability of a success (p) and the probability of a failure (1 - p) always add up to 1, or 100 percent.

Mean

The expected value, also known as the mean of the Bernoulli distribution, is represented as:

E(X) = p

This makes sense because if success has a probability of p and you assign it a value of “1”, then the average over time will be p. For example, let’s say you make a basket with a probability of 30 percent (p = 0.3). If you expect to get a success (basket = 1) 30 percent of the time and a failure (no basket = 0) 70 percent of the time, the combined ones and zeros will average to be 0.3, or p, after many attempts.

Variance

The variance represents how spread out the values are around the mean. The equation is: Var(X) = p(1 - p) = pq

The variance is largest when the probability is 0.5, because it is truly random and you can’t predict the result. As the probability moves toward 0 or 1, you have a better chance of predicting the outcome.

Standard deviation

The standard deviation is the square root of the variance and converts the value back into its original units.

SD(X) = sqrt(p(1 - p))

So consider you have a coin with p=0.5, meaning you have a 50 percent chance of getting either heads or tails when you flip it. The standard deviation here would be sqrt(0.5 * 0.5) = 0.5.

If you go back to the basket example, then the standard deviation would be sqrt(0.3 * 0.7) = 0.45. So you see, the standard deviation becomes smaller as you increase the reliability.

Examples of Bernoulli distributions

You can find Bernoulli distributions anywhere the outcome has two clear options. Some ways you might see it in practice include:

• Coin flip (heads or tails)

• Passing a class (pass or fail)

• An image contains a person (yes or no)

• An athlete scores a goal (yes or no)

• A salesperson meets their quota (yes or no)

When to use the Bernoulli distribution

As a quick check on whether to use the Bernoulli distribution, consider these questions:

• Does the event have two clear outcomes?

• Is the probability of success the same for each trial?

• Are you looking at the results of a single trial rather than many?

If you answer yes to all three questions, you can use a Bernoulli distribution.

Applications of the Bernoulli distribution across industries

Professionals use Bernoulli distributions when assessing clear binary outcomes. For example, businesses could measure the probability that a customer will click, subscribe, or purchase a product. This customer action is a binary yes/no event.

In machine learning, Bernoulli variables appear in binary classification problems, where the predictions have a clear success-or-failure outcome. Other applications include assessing whether a treatment is successful or whether a product is defective, helping professionals model outcome likelihoods and determine their next steps.

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