Question 8: In a certain country, the probability that a woman reaches the age of 70 is 0.7 and the probability that a woman lives to be 80 is 0.5. (a) If a woman from that country is 70 years old, what is the conditional probability that she will survive to 80 years? (b) Suppose that three sisters born in that country live completely independent lives. What is the probability that they will all be alive at the age of 70? (c) If the three sisters are all alive at the age of 70, what is the probability that they will all be alive at the age of 80?

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