Flip a coin 10 times — how many heads will come up? It might be exactly 5, it might be only 3, and if you're lucky, all 10 could come up heads. The probability distribution that deals with this question — when you independently repeat the same trial a set number of times, how many times does the desired result (success) come up — is exactly the binomial distribution.
A few conditions need to hold for the binomial distribution to apply. First, each trial's result must be either "success" or "failure" — one or the other. Second, the probability of success p must be exactly the same on every trial. Third, the trials must be independent, meaning they don't affect each other. Flipping a coin (the probability of heads is always 1/2), basketball free throws (assuming the same success rate every time), and inspecting parts (assuming a constant defect rate) all satisfy these conditions, so they can all be handled with a binomial distribution.
The probability of exactly k successes when a trial is repeated n times is written P(X=k) = ₙCₖ × pᵏ × (1−p)ⁿ⁻ᵏ. When this holds, we say the random variable X follows the binomial distribution B(n, p). Breaking down the formula: pᵏ(1−p)ⁿ⁻ᵏ is the probability of "one particular order" with k successes and (n−k) failures, and the combination ₙCₖ is the number of ways to choose which of the n slots are the successes. In other words, you find the total probability as "the probability of one case" × "the number of possible cases."
The mean (expected value) of a binomial distribution can be found very simply as E(X) = np. For example, flip a coin 10 times and you can expect 5 heads on average (10×0.5); take 10 free throws with a 70% success rate and you can expect 7 successes on average (10×0.7). The variance is V(X) = np(1−p), and the standard deviation is its square root.
The shape of the histogram changes dramatically depending on the value of p. If p is close to 0.5, it approaches an almost perfectly symmetric bell shape centered on the mean; the closer p gets to 0 or 1, the more the distribution piles up on one side, becoming asymmetric. For example, inspect 20 parts with a 5% defect rate, and the probability piles up near 0 or 1 defects, trailing off in a long tail to the right.
As n gets larger, the number of bars increases and the distribution gets smoother and smoother — and especially when p is close to 0.5, as n grows the shape approaches the familiar bell curve, the normal distribution. You can pick up this relationship next in the "Normal Distribution Lab."
On the activity page, you can choose a scenario like coin flipping, free throws, or parts defect inspection, or change n and p yourself with sliders, and watch in real time how the shape of the histogram and the position of the mean (np) change.