Probability
Probability
Allen Star Batch
Grade 12
Question:
If $A_1, A_2, \ldots, A_n$ be any events of the same sample space then:
$\sum_{i=1}^{n} P(A_i) = 1$
$\sum P(A_i) \leq 1$ if $A_1, A_2, \ldots, A_n$ are disjoint
$\sum P(A_i) \geq 1$ if $A_1, A_2, \ldots, A_n$ are exhaustive events
None of these
Step-by-Step Solution
Key Concept: For any events in a sample space: disjoint events satisfy P(A₁∪A₂∪...∪Aₙ) = Σ P(Aᵢ) ≤ 1; exhaustive events satisfy A₁∪A₂∪...∪Aₙ = S, so Σ P(Aᵢ) ≥ 1. Both conditions hold simultaneously only when events are both disjoint and exhaustive, giving Σ P(Aᵢ) = 1.
For events $A_1, A_2, \ldots, A_n$: (a) They may be overlapping, so the statement is false. (b) If disjoint and exhaustive, then $\sum P(A_i) = 1$; if only exclusive (disjoint), then $\sum P(A_i) \leq 1$; if exhaustive, then $\sum P(A_i) \geq 1$.
Correct Answer: 2,3