O Level MathematicsE8.5 Set notation for probability (use $\text{P}(A \cup B)$ and $\text{P}(A \cap B)$).

🎲 Union vs. Intersection: Mastering $\text{P}(A \cup B)$ & $\text{P}(A \cap B)$!

Edudent Academy
2 Feb 26

When two events can occur in the same experiment, we often need to find the probability of **either** event happening or **both** happening together. This is where the notations P(A∪B)P(A \cup B) (probability of the *union*) and P(A∩B)P(A \cap B) (probability of the *intersection*) become essential. Understanding these symbols helps you avoid double-counting outcomes and is frequently tested in O Level examinations.

Main Concept

The **Addition Rule** links the two ideas:
P(A∪B)=P(A)+P(B)−P(A∩B).P(A \cup B)=P(A)+P(B)-P(A \cap B).
• A∪BA \cup B means *A or B or both*; • A∩BA \cap B means *A and B at the same time*; • Subtracting P(A∩B)P(A \cap B) prevents counting the overlap twice. In exams, always sketch a quick Venn diagram—it clarifies which region each probability represents.

Worked Example: Venn Diagram in a Classroom

Problem: Out of 30 students, 18 like Mathematics (AA), 12 like Science (BB) and 7 like both subjects. Find (i) P(A∪B)P(A \cup B), (ii) the probability a student likes neither subject. Solution (step-by-step): 1. Total students =30=30. Convert numbers to probabilities by dividing by 30. 2. P(A)=1830=0.6P(A)=\frac{18}{30}=0.6, P(B)=1230=0.4P(B)=\frac{12}{30}=0.4, P(A∩B)=730≈0.233P(A \cap B)=\frac{7}{30}\approx0.233. 3. Apply the addition rule:
P(A∪B)=0.6+0.4−0.233=0.767.P(A \cup B)=0.6+0.4-0.233=0.767.
4. Probability of neither subject is the complement:
P(neither)=1−P(A∪B)=1−0.767=0.233.P(\text{neither})=1-P(A \cup B)=1-0.767=0.233.
Thus, there is a 76.7\% chance a student likes at least one of the two subjects, and a 23.3\% chance they like neither. Keep practising similar questions to build confidence!