Counts and trials: the binomial distribution

What you’ll learn

This is lesson 10 of Track 9 (Statistics & Probability for AI) and the close of Phase 3 (Random variables and the distributions that matter). After the continuous bell of the previous lesson, this one returns to discrete counts: how many successes happen in a fixed number of yes-or-no trials. You will learn when the binomial model applies, how to compute the chance of exactly k successes, the quick n-times-p shortcut for the expected count, and why this distribution sits under so much of how AI is measured. The source curriculum is Khan Academy’s Statistics & Probability course, by Sal Khan and the Khan Academy team, freely available and cited as further study.

The lesson lays out the four binomial conditions, builds the exactly-k formula from counting arrangements, works it on three coin flips (P of 2 heads = 3/8) and on a model’s accuracy (P of 4 correct out of 5 at 80% accuracy, about 0.41), gives the expected-count shortcut n times p, separates “exactly k” from “at least k” (with the complement trick for “at least one”), and closes on the binomial in AI: accuracy as a count of correct predictions, conversion rates, and the large-n bridge to the normal.

Where this fits

This is lesson 10 of 14 and the final lesson of Phase 3. It is a specific random variable, so it builds on the expected-value lesson, and it connects to the normal distribution from the previous lesson (a large-n binomial looks normal). It also sets up Phase 4: a model’s accuracy on a test set is a binomial count, and asking how much to trust that count is the inference work that begins with the next lesson, From sample to population: sampling and the central limit theorem.

Before you start

Prerequisites: Random variables and expected value (lesson 8), since the binomial is a particular random variable and uses its expected-value idea. The probability rules from Phase 2 (independence, the complement) also recur here. Comfort with multiplying a few decimals and small whole-number counting is all the math needed.

About the math

The arithmetic is a little heavier than earlier lessons but still hand-sized: raising a probability to a small power, multiplying, and a small counting number C(n, k) that the lesson introduces gently (no factorials required to follow the examples). Every calculation is worked step by step on a concrete example, and the coin example is double-checked by listing all outcomes.

By the end, you’ll be able to

Identify when the binomial setting applies (fixed trials, two outcomes, constant probability, independence)
Compute the probability of exactly k successes in n trials using the binomial formula
Use the n-times-p shortcut for the expected number of successes
Distinguish “exactly k” from “at least k” and know that the latter requires summing or a complement
Recognize the binomial behind counting successes in AI (correct predictions out of a test set, conversions out of visitors)

Time and difficulty

Read time: about 12 minutes
Practice time: about 16 minutes (a self-check, a compute-the-binomial-probability exercise, an is-it-binomial-and-at-least-one exercise, and flashcards)
Difficulty: standard (the heaviest arithmetic in the track so far, but every step is worked out)