Events A and B overlap, with P(A) = 0.5, P(B) = 0.4, and P(A ∩ B) = 0.2. What is P(A ∪ B)?
A0.9
B0.7
C0.3
D1.1
The general addition rule gives P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = 0.5 + 0.4 − 0.2 = 0.7. The overlap is subtracted because P(A) and P(B) each already include the probability of the intersection, so adding them together counts P(A ∩ B) twice.
Question 2 True / False
For any two events A and B, P(A ∪ B) = P(A) + P(B).
TTrue
FFalse
Answer: False
This formula only holds when A and B are disjoint (mutually exclusive) — i.e., P(A ∩ B) = 0. For overlapping events, the correct formula is P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Applying the simpler formula to overlapping events overcounts the shared outcomes and can even produce probabilities greater than 1.
Question 3 Short Answer
Using only the three Kolmogorov axioms, explain why P(∅) = 0.
Think about your answer, then reveal below.
Model answer: The sample space S and the empty set ∅ are disjoint (they share no outcomes), and their union is S. By axiom 3, P(S ∪ ∅) = P(S) + P(∅). Since S ∪ ∅ = S, we have P(S) = P(S) + P(∅), which forces P(∅) = 0.
This derivation shows that P(∅) = 0 is not a separate assumption — it follows from the three axioms. The key move is recognizing that S and ∅ satisfy the disjointness condition in axiom 3 (A ∩ B = ∅ is trivially true when one set is empty), allowing the additive rule to apply.