Question

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

Answer 1

Answer:

• The two events are independent.

Step-by-step 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,

$P(A\bigcap B)=P(A)\times P(B)$

Now we know that:

$P(A|B)=\dfrac{P(A\bigcap B)}{P(B)}$

Hence,

$P(A|B)=\dfrac{P(A)\times P(B)}{P(B)}\\\\i.e.\\\\P(A|B)=P(A)$ )

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:

$P(A)=\dfrac{36}{72}\\\\i.e.\\\\P(A)=\dfrac{1}{2}$

Also,

Region B covers two parts of the total area.

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

Hence, we have:

$P(B)=\dfrac{36}{72}\\\\i.e.\\\\P(B)=\dfrac{1}{2}$

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

i.e.

Area of A∩B=74/4=18

Hence, we have:

$P(A\bigcap B)=\dfrac{18}{72}\\\\i.e.\\\\P(A\bigcap B)=\dfrac{1}{4}$

Hence, we have:

$P(A|B)=\dfrac{\dfrac{1}{4}}{\dfrac{1}{2}}\\\\i.e.\\\\P(A|B)=\dfrac{2}{4}\\\\i.e.\\\\P(A|B)=\dfrac{1}{2}$

Hence, we have:

$P(A|B)=P(A)$

         Similarly we will have:

$P(B|A)=P(B)$