Answer:
read below
Explanation:
To determine which events are independent and which are disjoint, let's first define what independence and disjointness mean:
1. Independence: Events A and B are independent if the occurrence of one event does not affect the probability of the other event. Mathematically, if P(A ∩ B) = P(A) * P(B), then events A and B are independent.
2. Disjoint (orthogonal): Events A and B are disjoint if they cannot occur simultaneously, meaning their intersection is empty, i.e., A ∩ B = ∅.
Now, let's analyze the given events:
- Independence:
- Events A₁ and B₁ are independent if P(A₁ ∩ B₁) = P(A₁) * P(B₁).
- P(A₁ ∩ B₁) = P({1}) = 1/4
- P(A₁) = P({1, 2}) = P({1}) + P({2}) = 1/4 + 1/4 = 1/2
- P(B₁) = P({1, 3}) = P({1}) + P({3}) = 1/4 + 1/4 = 1/2
- P(A₁) * P(B₁) = (1/2) * (1/2) = 1/4
- Since P(A₁ ∩ B₁) = P(A₁) * P(B₁), events A₁ and B₁ are independent.
- Similarly, events A₂ and B₂ are also independent.
- Disjoint:
- Events A₁ and A₂ are disjoint if A₁ ∩ A₂ = ∅.
- A₁ ∩ A₂ = {1, 2} ∩ {3, 4} = ∅
- Therefore, events A₁ and A₂ are disjoint.
- Similarly, events B₁ and B₂ are disjoint.
So, to summarize:
- Events A₁ and B₁ are independent.
- Events A₂ and B₂ are independent.
- Events A₁ and A₂ are disjoint.
- Events B₁ and B₂ are disjoint.
These conclusions are verified based on the given probability measure and event definitions.
