Probability
Probability examines classical probability, independent and dependent events, conditional probability, and expected value. EJU Course 2 problems often use draws without replacement or two-stage experiments.
Core rules
- Classical probability = favorable equally likely outcomes / total outcomes.
- Addition: P(A∪B)=P(A)+P(B)-P(A∩B). Mutually exclusive events have P(A∩B)=0.
- Multiplication: P(A∩B)=P(A)P(B|A). Events are independent exactly when P(A∩B)=P(A)P(B).
- Conditional probability: P(B|A)=P(A∩B)/P(A), provided P(A)>0.
- Expected value: E(X)=Σ x_i P(X=x_i).
- For a two-stage experiment, a tree diagram can organize the conditional probabilities.
Worked example
A bag has 3 red and 2 blue balls. Draw two without replacement. Find P(second is red | first is blue). After a blue ball is removed, 3 red and 1 blue remain, so the conditional probability is 3/4. Unconditionally, P(first blue and second red)=(2/5)(3/4)=3/10.
Traps
Do not multiply P(A)P(B) for dependent draws. Do not confuse mutually exclusive with independent: mutually exclusive events with positive probabilities are not independent. In conditional probability, the denominator is the conditioning event, not the total sample space. For expected value, multiply each value by its probability and include all outcomes; probabilities must sum to 1.