Question

The Venn Diagram below models probabilities of three events, A,B, and C.

Answer 1

• The two events are independent.

Explanation:

By the conditional property we have:

If A and B are two events then A and B are independent if:

`P(A|B)=P(A)`

or

`P(B|A)=P(B)`

( since,

if two events A and B are independent then,

`P(A\bigcap B)=P(A)* P(B)`

Now we know that:

`P(A|B)=(P(A\bigcap B))/(P(B))`

Hence,

`P(A|B)=(P(A)* P(B))/(P(B))\\\\i.e.\\\\P(A|B)=P(A)` )

Based on the diagram that is given to us we observe that:

Region A covers two parts of the total area.

Hence, Area of Region A= 72/2=36

Hence, we have:

`P(A)=(36)/(72)\\\\i.e.\\\\P(A)=(1)/(2)`

Also,

Region B covers two parts of the total area.

Hence, Area of Region B= 72/2=36

Hence, we have:

`P(B)=(36)/(72)\\\\i.e.\\\\P(B)=(1)/(2)`

and A∩B covers one part of the total area.

i.e.

Area of A∩B=74/4=18

Hence, we have:

`P(A\bigcap B)=(18)/(72)\\\\i.e.\\\\P(A\bigcap B)=(1)/(4)`

Hence, we have:

`P(A|B)=((1)/(4))/((1)/(2))\\\\i.e.\\\\P(A|B)=(2)/(4)\\\\i.e.\\\\P(A|B)=(1)/(2)`

Hence, we have:

`P(A|B)=P(A)`

Similarly we will have:

`P(B|A)=P(B)`