Mathematics: Analysis & Approaches HL · Statistics & Probability · 15 min read · Updated 2026-05-11
Basic probability concepts and rules — IB Mathematics: Analysis and Approaches HL
IB Mathematics: Analysis and Approaches HL · Statistics & Probability · 15 min read
1. Core Probability Terminology★☆☆☆☆⏱ 4 min
2. Complementary Events and the Complement Rule★☆☆☆☆⏱ 3 min
The complement rule follows directly from the fact that $P(S) = 1$:
P(A) + P(A') = 1 \implies P(A) = 1 - P(A')
This rule is especially useful for calculating "at least one" probabilities, where calculating the complement is much simpler than calculating the event directly.
3. Addition Rule for Combined Events★★☆☆☆⏱ 4 min
The general addition rule calculates the probability that either $A$ or $B$ (or both) occur:
P(A \cup B) = P(A) + P(B) - P(A \cap B)
If $A$ and $B$ are mutually exclusive, this simplifies to $P(A \cup B) = P(A) + P(B)$, since $P(A \cap B) = 0$.
4. Independent Events and the Multiplication Rule★★☆☆☆⏱ 4 min
For independent events, the multiplication rule gives the probability that both events occur:
P(A \cap B) = P(A) \times P(B)
Common Pitfalls
Why: Mutually exclusive describes overlap of outcomes, while independent describes a probability relationship between events. These are unrelated concepts.
Why: If events are not mutually exclusive, overlapping outcomes are counted twice when you add $P(A)$ and $P(B)$.
Why: The rule $P(A \cap B) = P(A)P(B)$ only holds for independent events.
Why: Many random experiments do not have uniform probability for outcomes, so counting outcomes directly will give the wrong result.