Chapter 2 - Probability Review

A probability space defined by Kolmogorov consists of

• A set of outcomes $S$, e.g., for the roll of a die, $S=\{1,2,3,4,5,6\}$ for the roll of two dice, $S=\{(\genfrac{}{}{0ex}{}{1}{1}),(\genfrac{}{}{0ex}{}{1}{2}),\dots ,(\genfrac{}{}{0ex}{}{6}{6})\}$ temperature on Monday, $S=[-50,50]$
• A set of events, where an event is a Subset of $S$, e.g., roll at least one 3: $\{(\genfrac{}{}{0ex}{}{1}{3}),(\genfrac{}{}{0ex}{}{3}{1}),\dots ,(\genfrac{}{}{0ex}{}{3}{3})\}$ temperature above $80\mathrm{deg}$: $(80,150]$

The union, intersection and complement of events are also events (an algebra).

A probability measure, gives a number between $0$ and $1$ to each event, where:

1. $P(S)=1$
2. $\text{given}A\cap B=\varnothing \Rightarrow P(A\cup B)=P(A)+P(B)$