1. the table shows the probabilities of a response chocolate or vanilla when asking a child or adult. use the formula for conditional probability to determine independence. a. are the events “chocolate” and “adults” independent? why or why not? b. are the events “children” and “chocolate” independent? why or why not? c. are the events “vanilla” and “children” independent? why or why not? chocolate | vanilla | total children | 0.14 | 0.26 | 0.40 adults | 0.21 | 0.39 | 0.60 total | 0.35 | 0.65 | 1.00

If p(x) is the total value of the production when there are x workers in a plant, then the average productivity of the workforce at the plant is a(x) = p(x) x . (a) find a'(x). a'(x) = xp'(x) − p(x) x a'(x) = xp'(x) − p(x) x2 a'(x) = p'(x) − p(x) x a'(x) = xp'(x) − p'(x) x2 a'(x) = p'(x) − xp(x) x2 why does the company want to hire more workers if a'(x) > 0? a'(x) > 0 ⇒ a(x) is ; that is, the average productivity as the size of the workforce increases. (b) if p'(x) is greater than the average productivity, which of the following must be true? p'(x) − xp(x) > 0 p'(x) − xp(x) < 0 xp'(x) − p'(x) > 0 xp'(x) − p(x) < 0 xp'(x) − p(x) > 0