4 · IB Maths AA
Venn diagrams and conditional probability
Choose P(A), P(B) and P(A ∩ B) and watch the Venn diagram reshape. Shade any region, read off P(A | B), and test whether the events are independent or mutually exclusive.
P(A | B) = P(A ∩ B) ÷ P(B)
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Independent ⇔ P(A ∩ B) = P(A) × P(B)
Mutually exclusive ⇔ P(A ∩ B) = 0
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Independent ⇔ P(A ∩ B) = P(A) × P(B)
Mutually exclusive ⇔ P(A ∩ B) = 0
- P(A ∪ B)
- P(A | B)
- P(B | A)
- P(A) × P(B)
- P(A ∩ B′), P(A′ ∩ B)
- P(A′ ∩ B′)
Try this
- Set P(A) = 0.6 and P(B) = 0.5. Find the value of P(A ∩ B) that makes A and B independent, then check that P(A | B) equals P(A).
- Set P(A) = 0.7 and P(B) = 0.6. Why will the P(A ∩ B) slider not go below 0.30? Use P(A ∪ B) ≤ 1 to explain.
- Choose the mutually exclusive preset with “Given B” shading. What is P(A | B) now, and why does that prove the events cannot be independent?
Common error: treating “mutually exclusive” and “independent” as the same idea — if P(A) and P(B) are both non-zero, mutually exclusive events are always dependent, because P(A | B) = 0 ≠ P(A).