SOLUTION: A and B are the two events in S, and P(A)=0.9 and P(B)=0.6 a) Is it possible that P(A ∩ B) = 0.1? Why or why not? b) What is the smallest possible value for P(A ∩ B)?

Algebra ->  Probability-and-statistics -> SOLUTION: A and B are the two events in S, and P(A)=0.9 and P(B)=0.6 a) Is it possible that P(A ∩ B) = 0.1? Why or why not? b) What is the smallest possible value for P(A ∩ B)?       Log On


   

Question 1170103: A and B are the two events in S, and P(A)=0.9 and P(B)=0.6
a) Is it possible that P(A ∩ B) = 0.1? Why or why not?
b) What is the smallest possible value for P(A ∩ B)?
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
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We have  

    P(A U B) = P(A) + P(B) - P(A ∩ B) <= 1.


Hence,

    P(A ∩ B) >= P(A) + P(B) - 1 = 0.9 + 0.6 - 1 = 1.5 - 1 = 0.5.


Therefore

(a)  It is not possible that P(A ∩ B) = 0.1  (see above)


(b)  Smallest possible value for P(A ∩ B)  is  0.5 in this case


          (and  max(0, P(A)+P(B)-1)  in general case).

Solved, answered, explained and completed.