Mathematics
Bernoulli Trials
Quick fact
The term 'Bernoulli trial' honors Jacob Bernoulli, who studied these binary outcomes in the 17th century. The Bernoulli distribution is the probability distribution of a single Bernoulli trial.
Why this is interesting
We all know a coin flip is either heads or tails. But what if that simple flip could model customer purchases, disease tests, or even success in sports? How can such a basic idea unlock so much?
Read the full explanation
Understanding Bernoulli Trials
Imagine you flip a fair coin. There are only two possible results: heads or tails. Each flip is independent — the result of one doesn't influence the next. This is a perfect example of a Bernoulli trial. In general, a Bernoulli trial is any experiment with exactly two outcomes. We call one 'success' (usually with probability p) and the other 'failure' (probability 1−p). The key idea is that the probability stays the same each time, and trials don't affect each other. For instance, if you test a light bulb to see if it works (success) or is broken (failure), that’s a Bernoulli trial. The outcome is binary, the chance of success is constant, and each test is separate. This concept becomes powerful when you repeat the same Bernoulli trial many times — then you can count successes and analyze patterns.
A deeper explanation
The Bernoulli trial is the simplest probabilistic experiment. Underlying it is the Bernoulli distribution: a random variable X that takes value 1 (success) with probability p and 0 (failure) with probability 1−p. Its expected value is p, and variance is p(1−p). The independence condition is crucial: without it, the trials wouldn’t be Bernoulli. This independence allows us to combine multiple Bernoulli trials into a binomial experiment, where we count successes over n trials. Bernoulli trials appear everywhere: in quality control (defective vs. non-defective), in online A/B testing (click vs. no click), and in clinical trials (response vs. no response). The simplicity of the concept masks its power — it is the foundation for understanding randomness in binary outcomes, and it introduces ideas like probability distributions, expected value, and variance in a concrete way.