Paper 2, Section I, F

Probability | Part IA, 2011

What does it mean to say that events A1,,AnA_{1}, \ldots, A_{n} are (i) pairwise independent, (ii) independent?

Consider pairwise disjoint events B1,B2,B3B_{1}, B_{2}, B_{3} and CC, with

P(B1)=P(B2)=P(B3)=p and P(C)=q, where 3p+q1\mathbb{P}\left(B_{1}\right)=\mathbb{P}\left(B_{2}\right)=\mathbb{P}\left(B_{3}\right)=p \text { and } \mathbb{P}(C)=q, \text { where } 3 p+q \leqslant 1

Let 0q1/160 \leqslant q \leqslant 1 / 16. Prove that the events B1C,B2CB_{1} \cup C, B_{2} \cup C and B3CB_{3} \cup C are pairwise independent if and only if

p=q+q.p=-q+\sqrt{q} .

Prove or disprove that there exist p>0p>0 and q>0q>0 such that these three events are independent.

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