They are not two words for one idea. They answer two
different questions.
S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Mutually exclusive — can both happen at once?
A = {1,2,3,4,5}
C = {7, 9}
A is the low half. C is two of the high numbers.
P(A AND C) = 0
Mutually exclusive means the overlap is empty.
Change one set, and the answer changes
B = {4,5,6,7,8}
B is the middle five — it reaches into A.
A AND B = {4, 5}
=
2 ÷ 10
=
0.2
0.2 ≠ 0
→ not 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 condition → re-scale what is left
P(A) = 0.50
D = {4,5,6,7}
D is the middle four — it also reaches into A.
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) =
0.20 → they agree
Overlapping, like D — and yet
B = {4,5,6,7,8}
A AND B = {4, 5}
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) =
0.20 → they 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 exclusive →
P(A AND B) = 0
Independent →
P(A AND B) = P(A) × P(B)