On the overlap of two events.

Question

Suppose for events A and B, P(A) = 1/2, P(B) = 2/3, but we have no more information about
the events.
(a) What are the maximum and minimum possible values for P(A U B)?
(b) What are the maximum and minimum possible values for P(A|B)?

Answer 1

Note event A may be included inside event B, or they may just overlap.  The minimum value for P(A∪B) is 2/3 and occurs when event A is included inside event B.  The maximum value for P(A∪B) is 1, because probabilities cannot be greater than 1, and occurs when event A overlaps event B.

 

P(A|B) = P(A∩B) / P(B)

So you need to find P(A∩B).

P(A∩B) = P(A) + P(B) - P(A∪B)

Two cases as discussed above:  

P(A∩B) = 1/2 + 2/3 - 2/3 = 1/2   (when event A is included inside event B)

or

P(A∩B) = 1/2 + 2/3 - 1 = 1/6   (occurs when event A overlaps event B)

 

Using the two cases:

Maximum:  P(A|B) = P(A∩B) / P(B) = (1/2) / (2/3) = 3/4    

Minimum:  P(A|B) = P(A∩B) / P(B) = (1/6) / (2/3) = 1/4