# Density is not probability

A density is probability per unit length. A probability belongs to an interval, and it is the area under the curve over that interval.

Written for this edition, not translated from Einstein.

## What does the height of a probability curve mean?

Mark where 100 particles end up on a ruler divided into 2 μm bins. If 30 land in one bin, that bin holds a probability of about 0.3, and the density there is 0.3 divided by 2 μm, or 0.15 per micrometre. Measured in metres the same density is 150,000 per metre, while the bin still holds 0.3: the height depends on the unit of length, and the probability does not.

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

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

Under a smooth curve, any single exact position has probability zero, since an interval of zero width has no area. At the starting instant of the Brownian problem every particle sits at one point. No curve can draw that: the whole probability, 1, sits on the single point, and a curve appears only once the particles have begun to spread.

## Worked example: A flat density across four micrometres

1. Spread the probability evenly over 4 μm. The density is 1 divided by 4 μm, or 0.25 per micrometre.
2. The area over the whole 4 μm is 0.25 per μm × 4 μm = 1.
3. A 1 μm piece has probability 0.25 × 1 = 0.25, and a 2 μm piece has 0.5.
4. Squeeze the same probability into 2 μm and the height doubles to 0.5 per micrometre, while the total stays 1. The height was never a probability.

## Where this lesson stops

The height of the curve is a density, a probability per unit length. Only the area over an interval is a probability.

## This lesson builds on

- [Adding and averaging](/foundations/bridge-sum-average/)