Introduction to Probability

Quantifying Uncertainty

Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). It is the mathematics of uncertainty behind statistics and machine learning.

Key structural terms:

For equally likely outcomes, probability is a ratio: $$P(E) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$$ For a fair die, $P(\text{even}) = \frac{3}{6} = \frac{1}{2}$.

Rules and Pitfalls

Probability obeys axioms: $P(E) \ge 0$, $P(S) = 1$, and for mutually exclusive events (cannot both happen), probabilities add: $P(A \cup B) = P(A) + P(B)$.

From these come vital operational rules:

Common pitfall: Adding $P(A) + P(B)$ when events can overlap. Simple addition is only for mutually exclusive events; otherwise, you must subtract the intersection or risk exceeding 1.