If AcB then what's the Pr[AuB]

Question

Assume Pr(A)=.15, Pr(B)=.25 Find Pr[AuB] in the following conditions...

 

1) if Ac(subset)B then Pr[AuB]

2) if AnB=0 then Pr[AuB]

3) if AnB'=0 then Pr[AuB]

Answer 1

It's often helpful to draw a Venn Diagram for questions like these.

 

 

 

(1)

 

If A is fully contained in B, then the probability of A or B occurring is the same as the probability of B occurring. Picture this with a Venn Diagram: Draw a circle for B and a circle entirely within that circle for A. Then label the A circle 0.15. All of B is 0.25, so label the part of B outside A 0.1.

 

Together, the two circles make up 0.1 + 0.15 = 0.25, which is just the probability of B. So:

 

P(A∪B) = 0.25

 

 

 

(2)

 

If the sets' intersection is empty, then no part of A is contained in B or vice versa. So, draw two circles that don't overlap at all. The first, A, has a probability of 0.15. The second, B, has a probability of 0.25.

 

Together, the two circles make up 0.15 + 0.25 = 0.4. So:

 

P(A∪B) = 0.4

 

 

 

(3)

 

If set A's intersection with B' is empty, then there are no elements in set A that are not in set B. That is, we're back in the first situation: A is fully contained in B. So:

 

P(A∪B) = 0.25