## Probability

### The Analysis of Data, volume 1

Basic Definitions: The Classical Probability Model on Continuous Spaces

## 1.4. The Classical Probability Model on Continuous Spaces

For a continuous sample space of dimension $n$ (for example $\Omega=\R^n$), we define the classical probability function as $\P(A)=\frac{\text{vol}_n(A)}{\text{vol}_n(\Omega)},$ where $\text{vol}_n(S)$ is the $n$-dimensional volume of the set $S$. The 1-dimensional volume of a set $S\subset \mathbb{R}$ is its length. The 2-dimensional volume of a set $S\subset \mathbb{R}^2$ is its area. The 3-dimensional volume of a set $S\subset \mathbb{R}^3$ is its volume. In general, the $n$-dimensional volume of $A$ is the $n$-dimensional integral of the constant function 1 over the set $A$.

Example 1.4.1. In an experiment measuring the weight of residents in a particular geographical region, the sample space could be $\Omega=(0,1000)\subset \mathbb{R}^1$ (assuming our measurement units are pounds and people weigh less than 1000 pounds). The probability of getting a measurement between 150 and 250 (in the classical model) is the ratio of the 1-dimensional volumes or lengths: $\P((150,250))=\frac{|250-150|}{|1000-0|}=0.1.$ The classical model in this case is highly inaccurate and not likely to be useful.
Example 1.4.2. Assuming the classical model on the sample space of Example 1.1.2, the probability of hitting the bullseye is $\P\left(\left\{(x,y)\,:\,\sqrt{x^2+y^2} < 0.1\right\}\right)=\frac{\pi \,0.1^2}{\pi \,1^2}=0.01$ (since the area of a circle of radius $r$ is $\pi\cdot r^2$). The classical model in this case assumes that the person throwing the darts does not make any attempt to hit the center. For most dart throwers this model is inaccurate.

• For the classical model to apply, the sample space $\Omega$ must by finite or be continuous with a finite non-zero volume.
• The classical model (on both finite and continuous spaces) satisfies the three axioms defining a probability function.
• A consequence of the classical model on continuous spaces is that the probability of an elementary event is zero (the volume of a single element is 0).
• In the next two chapters we will explore a number of alternative probability models that may be more accurate than the classical model.