## Probability

### The Analysis of Data, volume 1

Basic Definitions: The Classical Probability Model on Continuous Spaces
$
\def\P{\mathsf{P}}
\def\R{\mathbb{R}}
$

## 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.