Question 59039
The company feels that it has a 60% chance of winning contract A, and a 50% chance of winning contract B. Furthermore, the company believes that it has an 80% chance of winning contract A given that it wins contract B.
P(A)=0.6; P(not A)=0.4
P(B)=0.5; P(not B)=0.5
P(A|B)=0.8; P(not A|B)=0.2
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a) What is the probability that the company will win both contracts? 
P(A and B)= P(A|B)*P(B)=0.8*0.5=0.4
Comment: 
Notice that P(A and B) is not equal to P(A)*P(B)
because 0.4 is not equal to 0.6*0.5=0.3
Therefore events A and B are dependent.

b) What is the probability that the company will win at least one of the two contracts?
P(at least one)= 1 - P(win none)= 1- 0.4*0.5=0.8 

c) If the company wins contract B, what is the probability that it will not win contract A? 
P(notA |B)= P(not A and B)/P(B) = 0.4*0.5/0.5 = 0.4

d) What is the probability that the company will win at most one of the two contracts? 
P(at most one) = 1 - P(both)= 1-0.4 = 0.6

e) What is the probability that the company will win neither contract?
P(not A and notB)=0.4*0.5 = 0.2
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Cheers,
Stan H.