Bernoulli trial

A Bernoulli trial is a single trial of a chance process with exactly two outcomes, one labeled success and the other failure, and a success probability $p$ that is fixed for that trial. Code it as a random variable taking 1 for success and 0 for failure and it becomes the smallest random variable worth studying, with $\mu_X = p$ and $\sigma_X = \sqrt{p(1-p)}$.

One free throw from a 78 percent shooter is a Bernoulli trial with $p = 0.78$. Its mean is 0.78 makes, which is a long-run share rather than anything a single shot can produce, and its standard deviation is $\sqrt{0.78(0.22)} = \sqrt{0.1716} \approx 0.414$.

"Every card I draw is red or black, so each draw is a Bernoulli trial with $p = 0.5$" is where this goes wrong. Two outcomes is not the whole condition. Deal one card and keep it, and the next draw has $p = 25/51 \approx 0.4902$ or $26/51 \approx 0.5098$ depending on what came first, so the draws no longer share a single $p$ and stacking them does not build a binomial count. Independence and a constant $p$ are what make trials interchangeable, and only the first draw is safe on its own.

There is no $n$ inside a Bernoulli trial. The moment a question asks how many successes in 20 attempts you have moved to the binomial, and the scaling is not the one people guess. Means add, so $\mu = 20(0.78) = 15.6$ makes. Standard deviations do not, so $\sigma = \sqrt{20(0.78)(0.22)} \approx 1.853$ rather than $20(0.414) = 8.28$. Variances are what add, and the square root then grows with $\sqrt{n}$.

No topic in the Fall 2026 AP Statistics course is named after the Bernoulli trial. The setting is stated inside topic 2.10 of Unit 2, The Binomial Distribution, so the conditions are what get tested and the name is worth carrying for later courses.