# Density is not probability

Authored explanation; editorial review pending.

Divide a ruler into bins. The fraction of observations in a bin estimates its probability. Dividing that fraction by the bin width estimates a density. Changing the ruler from metres to micrometres changes the numerical density but not the probability of the same physical interval.

$$
P(a\le X\le b)=\int_a^b p(x)\,dx
$$

The probability of the interval from a to b is the integral of the density over that interval.

A continuous distribution gives probability zero to any single exact position. At the starting time the ideal point source is instead an atom of probability one, not a finite curve with infinite height.

## Worked example

A uniform density of 0.25 per micrometre on a four-micrometre interval has total probability one. A one-micrometre subinterval has probability 0.25; a two-micrometre subinterval has probability 0.5.

Always distinguish curve height, interval area and an individual observation.