Question: If P(A) = 0.3, P(B) = 0.2, and P(A U B) = 0.46, then P(A intersect B) = . (a) Are events A and B independent? (enter YES or NO) (b) Are A and B mutually exclusive? (enter YES or NO)

 Solution:  Here we use the value you given to me 

P(A) = 0.3, P(B) = 0.2, P(A ∪ B)=0.46
so here we have to calculate the P(A ∩ B) :

P(A âˆ© B) =P(A) + P(B) − P(A ∪ B)
              = 0.3  +  0.2  -0.46
 P(A âˆ© B) =  0.04

So from this all values we show the below :

(a) Are events A and B independent?

        Two events are independent if the outcome of one event does not impact the outcome of the other event. That is, the two events are not related. There are three simple ways to check for independence using the probability rules :

1.  P(A)×P(B)=P(A∩B)
2. P(A∣B)=P(A)
3.  P(B∣A)=P(B)

So,here P(A)×P(B) = 0.3 * 0.2 = 0.06

            P(A)×P(B) ≠ P(A∩B)

So, here  events A and B are NOT independent.

(b) Are A and B mutually exclusive?

    Two events that do not occur at the same time,  known as mutually exclusive events

• A and B together is impossible: 
  i.e  P(A  âˆ© B) = 0
• A or B is the sum of A and B: 
  i.e  P(A U B) = P(A) + P(B)
   

So, here we observe above result is not satified the both condition of Mutually exclusive so here A and B are NOT Mutually Exclusive.