Need help in verifying that my reasoning is right on operations with probability sets

Question

If we have: P(A); P(B); P(A ∪ B);

And we need to compute: P(A ∩ B), P(A' ∪ B') and P(A' ∩ B).

Where ' indicates complement.

Is it correct way of doing it:

P(A ∩ B) = P(A) + P(B) - P(A ∪ B)

P(A') = 1 - P(A)

P(B') = 1 - P(B)

P(A ∪ B)' = 1 - P(A ∪ B)

P(A' ∪ B') = P(A') + P(B') - P(A ∪ B)'

P(A' ∩ B) = P(B) - P(A ∩ B)

That is what I got trying to solve this question using Van Diagram...

Answer 1

Yes, those look correct, though an approach using fewer steps to be based on the original three values could have given

• $P(A \cap B) = P(A)+P(B) - P(A \cup B)$

• $P(A' \cup B') = P(A \cap B)' = 1 - P(A)-P(B) + P(A \cup B)$

• $P(A' \cap B) = P(A \cup B) - P(A)$