Basic Concepts

A probability model consists of:

1. A sample space \( \Omega \),
2. A collection of events (subsets of \( \Omega \)), and
3. A function \( P \), called the probability measure, that assigning probabilities to events.

The function \( P \) maps each event to a number in the interval \([0, 1]\) (real numbers from $0$ to $1$, inclusively), where \( P(\Omega) = 1 \).

In the uniform model, all outcomes are equally likely, and the probability of any event is computed by counting favorable and total outcomes. In more general, real-life models, however, probabilities can differ from outcome to outcome; for example, in a biased die, some faces might occur more often than others.

We use these models to predict the likelihood of events and simulate randomness in a mathematically rigorous way.