SOLUTION: A and B are two events in a sample space. Given that P(A)=0.4 and P(A or B)=0.7. Find the probability that neither A nor B occurs. Find the value of P(B) for which A and B are mu

(a)  The probability that neither A nor B occurs is the COMPLEMENT
     to the probability P(A or B).  

     For P(A or B), it is given:  P(A or B) = 0.7.

     THEREFORE, P(neither A nor B) = 1 - P(A or B) = 1 - 0.7 = 0.3.    It is the ANSWER to question (a).



(b)  If A and B are mutually exclusive, it means that they are disjoint. i.e. P(A and B) = 0.

     It implies that P(A or B) = P(A) + P(B), 
     
     and, substituting the given data, we have  0.7 = 0.4 + P(B),

     which implies  P(B) = 0.7 - 0.4 = 0.3.            It is the ANSWER to question (b).



(c)  A and B are independent (by the definition) if and only if P(A and B) = P(A)*P(B).

     Next, we use the general formula  P(A or B) = P(A) + P(B) - P(A and B)
     and substitute there  P(A and B) = P(A)*P(B).  It gives us

         P(A or B) = P(A) + P(B) - P(A)*P(B).


     Substitute here the given values. You will get

         0.7 = 0.4 + P(B) - 0.4*P(B).


      Simplify and get P(B)

         0.7 - 0.4 = P(B) - 0.4*P(B)

            0.3    =    0.6*P(B)

            P(B) =  =  = 0.5.    It is the ANSWER to question (c).

(1)  General formula of probability P(A or B) = P(A) + P(B) - P(A and B),
     which is valid ALWAYS for any events A and B.



(2)  Formula for independent events A and B  P(A and B) = P(A)*P(B).



(3)  Definition of mutually exclusive(= disjoint) events  P(A and B) = 0.


     Formula for probability of mutually exclusive (= disjoint) events   

         P(A or B) = P(A) + P(B).



(4)  Formula for P(neither A nor B)

         P(neither A nor B) = 1 - P(A or B).



(5)  General formula for probability P(nor A) = 1 - P(A)  

     (so called complementary probability).