a) Suppose A1,A2,A3 are events such that P(A1 ∩ A2) = P(A1 ∩ A3)!= 0 while P(A2 ∩ A3) = 0. Calculate the probability that at least one of these events occurs by calculating
P(A1 ∪ A2 ∪ A3) in terms of only P(A1),P(A2),P(A3) and P(A1 ∩ A2).
b) If A1, A2, A3 are pairwise independent events is it possible that P (A1 ∩ A2) = P (A1 ∩ A3) ̸= 0 while P (A2 ∩ A3) = 0, and what does it say about the events A1, A2, A3? You have to justify your answer with a calculation.
a)I have been trying to follow the Additive and Multiplication rule for two events, but I am not sure on how to isolate the independent event P(A2 ∩ A3) = 0.
What I did: P(A1 ∪ A2 ∪ A3)=P(A1 U A2)+P(A1 U A3)+P(A2 U A3)
Some help I have been trying to follow the method discussed it his exchange but I am stuck:
b) I understand that A1 has an intersection with A2 and A3, but A2 and A3 are distinct. I can show it in a Venn diagram, but I do not know how to prove it with a computation.
Any help would be appreciated.Thank You!`
