Suppose the following probabilities are known. p(a) = 0.40, p(b a) = 0.25, p(c | a) = 0.45, and p(d a) = 0.30 a find p(a n b), p(an c), and p(an d). p(a n b) p(a n c) p(a n d) are the events a and b independent? justify your answer. (b) independent, because p(a n b) = p(a) p(b) yes, they are independent, because the sum of the given probabilities is greater than 1 yes, they are no, they are dependent, because p(b | a) # p(b). no, they are dependent, because the sum of the given probabilities is greater than 1 there is not enough information to determine independence or dependence. find p(b' | a) (c) p(b' | a) if the event a occurs, are there any other events in addition to b, c, and d that can occur? justify your answer. if the event a occurs, other events in addition to b, c, and d can occur, because p(b a)p(c| a) p(d | a) = 1 if the event a occurs, other events in addition to b, c, and d can occur, because p(c i a) p(b i a) p(d i a) 1. p(d a) 1 if the event a occurs, only b, c, or d can occur, because p(b | a) p(c a) + p(c | a) p(d a) if the event a occurs, only b, c, or d can occur, because p(b | a) #1 + + there is not enough information to determine.