Bernoulli Distribution

TL;DR

• Models a single binary outcome using a single parameter p (probability of success).
• Probability mass function: $P(X=x)=p^x q^{1-x}$ with (q=1-p) and (x\in{0,1}).
• Assumes a constant success probability and independent outcomes, which can be limiting in real situations.

Definition

The Bernoulli distribution is a probability distribution for a binary experiment with two possible outcomes: success or failure. It is defined by the probability of success (p) and the probability of failure (q=1-p). For a random variable (X) representing the outcome and (x\in{0,1}),

$$P(X=x) = p^x q^{1-x}$$

The distribution is named after Jacob Bernoulli, introduced in Ars Conjectandi (1713).

Explanation

A Bernoulli-distributed variable represents a single trial with two outcomes: success (usually coded as 1) and failure (coded as 0). The model assigns probability (p) to success and (1-p) to failure. The closed-form expression (p^x q^{1-x}) yields the probability of the specific observed outcome (x). The distribution is commonly used when only two outcomes are possible and when the trial’s success probability is treated as constant and independent from other trials.

Examples

Coin toss

Consider a coin toss where the probability of landing on heads is 0.5. Using the Bernoulli distribution, the source gives:

P(X=1) = 0.5^1 * 0.5^(1-1) = 0.25

P(X=0) = 0.5^0 * 0.5^(1-0) = 0.25

Customer purchase

Consider the probability of a customer purchasing a product being 0.7. Using the Bernoulli distribution, the source gives:

P(X=1) = 0.7^1 * 0.3^(1-1) = 0.21

P(X=0) = 0.7^0 * 0.3^(1-0) = 0.79

Use cases

• Medical trials with binary outcomes (success or failure).
• Political polls with support versus opposition responses.
• Machine learning algorithms predicting the probability of an event occurring or not.

Notes or pitfalls

• Assumes the probability of success (p) is constant across trials.
• Assumes outcomes are independent; the result of one event does not affect another. These assumptions may not hold in complex real-world situations.

Related terms

• Binary experiment
• Probability distribution
• Success and failure