The events A and B are such that P(A)= 1/3 and P(B)=1/4. P(A or B or both)= 1/2.

Question

The events A and B are such that P(A)= 1/3 and P(B)=1/4. P(A or B or both)= 1/2. Represent these probabilities on a vent diagram. Show that A and B are independent

Answer 1

P(A and B) = P(A∩B)

 

A and B are independent when P(A∩B) = P(A)P(B)

 

P(A only) = 1/3 - P(A∩B)

 

P(B only) = 1/4 - P(A∩B)

 

P(A or B or both) = P(A∪B) = P(A only) + P(A∩B) + P(B only) = 1/2

 

So, [1/3 - P(A∩B)] + [1/4 - P(A∩B)] + P(A∩B) = 1/2

 

We get P(A∩B) = 1/3 + 1/4 - 1/2 = 1/12

 

P(A)P(B) = (1/3)(1/4) = 1/12

 

Since P(A∩B) = P(A)P(B), the events A and B are independent.

 

NOTE:  I solved the problem without using a VENN Diagram.  The Venn Diagram would be 2 intersecting circles, one circle representing set A and the other representing set B.  The intersection represents A∩B, the portion of circle A that doesn't include A∩B represents A only, and the portion of circle B not including A∩B represents B only.