Find p(A), pB)andp(C) .

Question

A1a. For three events A, B, and C, we know that · A and C are independent, · B and C are independent, · A and B are disjoint, · p(AorC)=2/3,p(BorC)=3/4 and p(AorBorC)=11/12. Find p(A),P(B)andp(C)?

Answer 1

Write "|" (vertical bar) to mean "or", so p(A|C)=p(A)+p(C)-p(A)p(C), which means the sum of the independent probabilities less the probability that both occur. Similarly, p(B|C)=p(B)+p(C)-p(B)p(C). And p(A|B|C)=p(A)+p(B|C)-p(A)p(B|C)=p(A)+p(B)+p(C)-p(B)p(C)-p(A)(p(B)+p(C)-p(B)p(C)). Now we can plug in figures: 2/3=p(A)+p(C)-p(A)p(C); 3/4=p(B)+p(C)-p(B)p(C); 11/12=p(A)+3/4-p(A)*3/4; 11/12-3/4=1/6=p(A)(1-3/4), p(A)=4/6=2/3. 2/3=2/3+p(C)-(2/3)p(C), p(C)=0 and p(B)=3/4. (We don't consider p(A|B|C) as p(A|B)+p(C)-p(A|B)p(C) because A and B are disjoint.)

Solution appears to be: p(A)=2/3, p(B)=3/4 and p(C)=0.