Question

Consider a situation in which P(A) = , P(C) = , and P(A and B) = . What is P(B and C)?

Answer 1

I can’t compute a numeric value because the probabilities were left out of your statement. Also, even if you give P(A), P(C) and P(A ∩ B) alone, P(B ∩ C) is not determined without more information about B (or relationships among the events).

What I can say in general:
- Without further assumptions, P(B ∩ C) can range from 0 up to min(P(B), P(C)).
- From P(A ∩ B) we only know P(B) ≥ P(A ∩ B), so the weakest bounds using the given symbols a = P(A), c = P(C), d = P(A ∩ B) are
- 0 ≤ P(B ∩ C) ≤ c.
So any value between 0 and P(C) is possible consistent with a, c, d.

If you add assumptions you can get a definite value. For example, if A and B are independent so P(B) = P(A ∩ B)/P(A) (assuming P(A) > 0) and also B and C are independent, then
P(B ∩ C) = P(B)P(C) = [P(A ∩ B)/P(A)] · P(C).

Give the missing numbers or any independence/relationship assumptions and I’ll compute the corresponding value or tighter bounds.