Bernoulli Trials

College/University

Definition

A trial in probability in which a random experiment with precisely two possible outcomes are repeated and it consistently gives the same probabilty of success. The binomial probability formula can help you find probabilites for Bernoulli trials.

Worked examples

\(P(\)3 heads in 5 flips\() = \binom{5}{3}(0.5)^3(0.5)^2 = 10 \cdot 0.03125 = 0.3125\)

Each flip is a Bernoulli trial (heads/tails, constant \(p=0.5\)); the binomial formula counts exactly 3 successes in 5 trials.

\(P(\)2 sixes in 4 rolls\() = \binom{4}{2}\left(\frac{1}{6}\right)^2\left(\frac{5}{6}\right)^2 \approx 0.1157\)

Rolling a die is a Bernoulli trial when you define success as rolling a six; probability stays \(\frac{1}{6}\) every roll.

Common mistakes

Using Bernoulli trials when the probability changes between trials → Bernoulli trials require the same probability of success every time Drawing cards without replacement changes probabilities, so those are not Bernoulli trials.
Thinking a trial with three outcomes (win/lose/draw) is Bernoulli → Bernoulli trials have exactly two outcomes You can redefine outcomes as success/failure (e.g., win vs not-win) to make it Bernoulli.
Assuming trials must be independent if there are two outcomes → Bernoulli trials must be independent and have constant probability Two outcomes alone is not enough; each trial must not affect the next.

Where you'll use it next

Bernoulli trials underpin binomial distributions, hypothesis testing, and confidence intervals in statistics. You'll also see them in geometric and negative binomial models when counting trials until a success.

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