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Introduction to Statistics
Independent or mutually exclusive?
Section 3.2 · Probability Topics
Introduction to Statistics · Section 3.2 Independent or mutually exclusive?
Two ideas that are frequently interchanged — and they are not two names for the same thing.
They are not two words for one idea. They answer two different questions.
Pick one of ten classmates at random
S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
One classmate per outcome — each equally likely to be the one you pick.
Mutually exclusive — can both happen at once?
= {1,2,3,4,5}A — took the bus
= {7, 9}C — walked
No one both rode the bus and walked. And 6, 8 and 10 did neither — so C is not simply “not A”.
P(A AND C) = 0
Mutually exclusive means the overlap is empty.
Change one set, and the answer changes
= {4,5,6,7,8}B — has a driver’s licence
You can hold a licence and still ride the bus — 4 and 5 did both.
A AND B = {4, 5} = 2 ÷ 10 = 0.2
0.2 = 0not mutually exclusive
Independent — does knowing one change the other?
P(A | B) = P(A)
Learning that B happened moves your estimate of A not at all.
What “given” actually does
delete every outcome outside the conditionre-scale what is left
P(A) = 0.50
= {4,5,6,7}D — has a part-time job
D reaches into A exactly as B does — students 4 and 5.
P(A | D) = 0.50 = P(A)
The world shrank. Your belief about A did not move. That is independence.
The same test, in arithmetic
P(A) × P(D) = 0.50 × 0.40 = 0.20
P(A AND D) = 0.20they agree
Overlapping, like D — and yet
B = {4,5,6,7,8}back to B — has a licence
A AND B = {4, 5}bussed and licensed
P(A | B) = 0.40 = P(A)
The bar moved. B is not independent of A.
P(A) × P(B) = 0.50 × 0.50 = 0.25
P(A AND B) = 0.20they disagree
Same overlap as D, opposite answer. Overlap alone decides nothing.
Same operation, different set
P(A | C) = 0
Your belief about A went from a half to nothing.
bar unchanged vs bar destroyed
Not neighbours on a scale. The two opposite ends of it.
Two questions, not one
overlap empty? vs does knowing change it?
Mutually exclusive events are the least independent events there are.
One test each
Mutually exclusiveP(A AND B) = 0
IndependentP(A AND B) = P(A) × P(B)
A — took the bus
C — walked
  • Ten classmates, one picked at random — every claim here is countable.
  • A, bussed and C, walked — the bands never share a student.
  • A, bussed and B, licensed — both bands sit under students 4 and 5.
  • One band now — A, took the bus. The bar is its share of all ten.
  • Given D, a part-time job: four students left, two of them bussed — still half.
  • Given B, a licence: five students left, two of them bussed — no longer half.
  • Given C, walked: two students left, and neither took the bus.

Key ideas

  • Mutually exclusive P(A AND B) = 0They cannot both happen.
  • Default assumption Assume not mutually exclusiveuntil you check.
  • Independent P(A | D) = P(A)Knowing D changes nothing.
  • In arithmetic P(A AND D) = P(A) × P(D)Multiply, then compare.
  • Overlap decides nothing D and B share the same twooutcomes — only one is independent.
  • The trap Mutually exclusive events arenever independent.