Lesson 4 of 11 · 7 min
Mutually exclusive and exhaustive events
NCERT §14.1.4, §14.1.5
The coin stall gives three kinds of prize: nothing for no head, a sticker for exactly one head, a sweet for two or more heads. The treasurer worries that one toss might earn two prizes, or none at all. Is that possible?
The lesson in notes
In short
A and B are mutually exclusive when they cannot happen together: A ∩ B = φ, so the sets are disjoint.
On a die, odd {1, 3, 5} and even {2, 4, 6} are mutually exclusive. Odd {1, 3, 5} and 'less than 4' {1, 2, 3} are not, because 3 lies in both.
Two different simple events can never happen together, so any two of them are mutually exclusive.
Events E₁, E₂, …, Eₙ are exhaustive when E₁ ∪ E₂ ∪ … ∪ Eₙ = S: whatever happens, at least one of them occurs. On a die, {1, 2, 3}, {3, 4} and {5, 6} are exhaustive, though the first two overlap.
If the events are also pairwise disjoint (Eᵢ ∩ Eⱼ = φ for i ≠ j), they are mutually exclusive and exhaustive: exactly one of them occurs every time.
NCERT's Example 2: two dice, with A 'sum even', B 'sum a multiple of 3', C 'sum less than 4', D 'sum greater than 11'. C holds only (1,1), (1,2), (2,1) and D only (6,6), so C and D are mutually exclusive; every other pair shares at least one outcome.
NCERT's Example 3: three coins with A 'no head', B 'exactly one head', C 'at least two heads'. They have 1, 3 and 4 points, never overlap and together fill S, so they are mutually exclusive and exhaustive.