$P(A) = \frac{|A|}{|S|}, \quad 0 \leq P(A) \leq 1$ $\text{where } S = \text{the sample space of all possible outcomes}$ Probability ($P$) is a numerical measure between 0 and 1 assigned to an event $A$ that quantifies the likelihood of that event occurring. A valid probability measure satisfies three axioms: $P(A) \geq 0$ for all events $A$; $P(S) = 1$; and for mutually exclusive events, $P(A \cup B) = P(A) + P(B)$. > [!example]- Drawing a Card from a Standard Deck {💡 Example} > A card is drawn at random from a standard 52-card deck. What is the probability it is a heart or a face card? > > > [!answer]- Answer > > Let $H$ = heart (13 cards), $F$ = face card (12 cards). Their intersection $H \cap F$ = face cards that are hearts = 3 cards. By inclusion-exclusion: > > $P(H \cup F) = P(H) + P(F) - P(H \cap F) = \frac{13}{52} + \frac{12}{52} - \frac{3}{52} = \frac{22}{52} \approx 0.423$