Probability measures how likely an event is, from 0 (impossible) to 1 (certain).

Fundamentals

\[ P(E)=\frac{\text{number of favorable outcomes}}{\text{number of equally likely outcomes}}. \]

Also: \(0\le P(E)\le1\).

Complement rule

\[ P(E')=1-P(E). \]

Combined events

• \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
• If independent: \(P(A\cap B)=P(A)P(B)\)

Experimental probability

\[ P(E)\approx\frac{\text{observed frequency}}{\text{number of trials}}. \]

Sample spaces and trees

Use sample-space tables or tree diagrams for multi-step outcomes. Multiply along branches, add across valid branches.

Exam strategy

• Define event clearly before calculating.
• Use fractions first; round at end if needed.
• Check complement and total probability sanity.
• Use diagrams to avoid missing outcomes.