Lesson 5 of 10 · 7 min
Partition of a sample space
NCERT §13.5.1
Rainy or dry is one way to sort the mornings. Asha now sorts them three ways. What makes a sorting usable?
The lesson in notes
In short
Events E₁, E₂, …, Eₙ form a partition of S if (a) no two of them overlap: Eᵢ ∩ Eⱼ = φ for i ≠ j; (b) together they cover S: E₁ ∪ E₂ ∪ … ∪ Eₙ = S; and (c) each has P(Eᵢ) > 0.
In words: pairwise disjoint, exhaustive, and each with non-zero probability. Exactly one of the Eᵢ occurs in any trial.
The probabilities of the parts add to 1.
Simplest partition: any event E (not empty, not all of S) together with its complement E′.
For any two events E and F, the four pieces E ∩ F, E ∩ F′, E′ ∩ F and E′ ∩ F′ form a partition of S.
A sample space has many partitions. For a die, {odd, even} is one and {1, 2}, {3, 4}, {5, 6} is another; choose the one that matches the cases in the problem.