An online bookseller uses one of four shipping companies to send packages to its
ID: 3086902 • Letter: A
Question
An online bookseller uses one of four shipping companies to send packages to its customers. Any package can be sent with one and only one of these companies. Define the following events: 1)C1: the package is shipped with company 1 2)C2: the package is shipped with company 2 3)C3: the package is shipped with company 3 4)C4: the package is shipped with company 4 The bookseller uses the shipping companies with the following probabilities: P(C1) = 0.5; P(C2) = 0.25; P(C3) = 0.125; P(C4) = 0.125. Let X be the event that the package arrives on time at its destination. Depending on the shipping company used, the probability of X varies: P(X|C1) = 0.85; P(X|C2) = 0.9; P(X|C3) = 0.8; P(X|C4) = 0.8. (a) Compute the numerical value of P(C2 U C3 ) . (b) Given that a package has arrived on time what is the probability that it was shipped with company C1? (c) Are the events C' ( it is the complement of C ) andX 11 independent? Justify your answer.Explanation / Answer
a) P(C2 U C3) = 0.25 + 0.125 - 0.25 x 0.125 = 0.34375
b) required P (C1| X). We use Baye's theorem
P (C1| X) = P(X|C1). P(C1)/P(L)
= 0.85x0.5/(0.85x0.5 + 0.9x0.25 + 0.8x0.125 + 0.8x0.125)
= 0.5
c) from the question, it is not clear what event C is. However, If the probability of the intersection of two events is zero, then the events are independent.