Binomial Distribution

Algebra 2 · Axiom Academy

Repeated independent success-or-failure trials — and the one formula that counts the successes. A binomial experiment repeats the same trial a fixed number of times, n . Every trial is independent and ends one of two ways — a success (probability p ) or a failure (probability 1-p ). We track only X , the number of successes. The trial is repeated a set number of times, n . Each trial is either a success or a failure — nothing else. One trial's result never changes the next one's. The success probability p is the same on every trial. To score exactly k successes in n trials, the successes can sit in different positions. Because the trials are independent, every such arrangement has the same probability p^ k (1-p)^ n-k . So the whole job is: count how many arrangements there are, then multiply. The count is a combination — how many ways to choose which trials are the successes. The animation lays out every way to place 2 successes among 4 trials. 3. The Shape of the Distribution Compute P(X=k) for every k from 0 to n and the values trace out a shape. It climbs to a peak near the mean and — when p=0.5 — is perfectly symmetric . Here is the fair case n=6 , p=0.5 . 4. The Mean and a Worked Probability On average, a binomial variable lands at E(X)=np . Take a player who makes free throws with probability p=0.8 and shoots n=5 of them: how many will she make on average, and how likely is it that she makes exactly 4 ? Play the animation to substitute and evaluate.

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