SOLUTION: p(A)=0.50,P(A/B)=O.50,P(B)=0.80. FIND p(A INTERSECTION B) AND P(A UNION B) IS 'A' AND 'B' ARE INDEPENDENT

Question 1142346: p(A)=0.50,P(A/B)=O.50,P(B)=0.80.
FIND p(A INTERSECTION B) AND P(A UNION B)
IS 'A' AND 'B' ARE INDEPENDENT
Answer by ikleyn(52908) (Show Source): You can put this solution on YOUR website!
.

By the definition,  P(A|B) = P(A & B)/P(B);


so, from the given data  


    P(A|B) = P(A & B)/P(B) = 0.50,


which gives  


    P(A & B) = 0.5*P(B) = 0.5*0.80 = 0.4.


Thus P(A intersection B) is just found:  it is  P(A intersection B) = P(A & B) = 0.4.


If  P(A & B) = 0.4,  then


    P(A U B) = P(A) + P(B) - P(A & B) = 0.5 + 0.8 - 0.4 = 0.9.


The last question is


        are A and B independent ?



Two events X and Y are called independent if P(X & Y) = P(X) * P(Y).


Lets check if it is valid for our A and B.


We just found that  P(A & B) = 0.4;  but  P(A)*P(B) = 0.5*0.4 = 0.2  has DIFFERENT value.



ANSWER.  P(A & B) = 0.4;  P(A U B) = 0.9;  

         the events A and B are NOT independent.

Solved.

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