Please help me to solve these problems

Question

1. Prove inequality: P(A∩B) ≥P(A)+P(B)−1. 2. Suppose p = P(A),q = P(B),r = P(A∪B), what are the probabilities P(A − B) and P(A∩B)?

Answer 1

1. We can rearrange the inequality such that:

P(A)+P(B)-P(A∩B)≤1

Note that the left term is equivalent to the probability of A or B occurring by definition, so that:

P(A)+P(B)-P(A∩B) = P(A∪B) ≤ 1

where this is true since the sum of all probabilities has to be less than or equal to 1.

1. 
   
2. P(A-B) can be interpreted as the probability of events in A but not in B. Intuitively, it is equivalent to P(A)-P(A∩B). Try to visualize it with a diagram if that helps. Can you solve it from there?