Binomial Distribution
Blast a signup offer to 200 inboxes, each person clicking or ignoring at the same 3% rate. You will rarely get exactly 6 clicks twice: the count wanders, and its entire spread rides on just two numbers.
- ▸Send the same offer twice and the click count shifts; two numbers, batch size and rate, fix its whole spread.
- ▸The average is just batch times rate: no summing, no simulation, one multiplication.
- ▸Grow the batch and the bars smooth into the very bell curve you already met.
Send a signup offer to ten inboxes. Each person does one of two things, clicks or ignores, and suppose everyone clicks at the same rate, say one in four. Run the campaign and you might get three clicks. Run the identical campaign tomorrow and you get two, or five. The number of clicks is not a fixed figure; it wanders across the whole range from zero to ten.
Each single inbox is one yes/no trial, a Bernoulli trial, and the campaign stacks ten of them. What you care about is the total: how many of the ten clicked. Notice that many different campaigns share a total. Exactly two clicks could be inboxes one and two, or one and seven, or nine and ten, and on and on. Every one of those distinct patterns lands the same total of two.
So a total is not one outcome but a bundle of them, and the size of the bundle is a plain count: how many click-patterns produce that exact total. That count has a name, the binomial coefficient, written C(10, 2) and read "ten choose two." Lay every possible total from zero to ten along an axis and you have the skeleton of a distribution: a row of bars, one per total, and each bar's raw material is the number of patterns that reach it.