1. Let A and B two events such that P( A) = 0.75, P(B) = 0.35, and P( A and B ) = 0.25 . (2 pts each) 3. Assume that the lengths of pregnancy for humans is approximately normally distributed, with a mean of 265 a. Find P(not A) [Show your work] b. Find P(not B) days and a standard deviation of 12 days. [Show your work] a. Draw a sketch of this normal distribution and label at least 5 points (one and two standard deviations from the mean) on the horizontal axis. (2 points) C. Find P(A or B) [Show your work] d. Find P(not (A or B)) [Show your work] e. Find P(A if B) [Show your work] f. Find P(B if A) b. What proportion of human pregnancies last longer than 280 days? (A labeled sketch of the distribution, [Show your work] corresponding shaded areas, and documented calculator or STATKEY commands are required for full credit. Box final answer and round to three decimal places past the decimal.) (3 points) Are the events disjoint (mutually exclusive)? Clearly answer "yes" or "no" and explain your answer. h. Are the events independent? Clearly answer "yes" or "no" and explain your answer. c. What proportion of human pregnancies last between 260 days and 280 days? (A labeled sketch of the distribution, corresponding shaded areas, and documented calculator commands are required for full credit. Box final answer and round to three decimal places past the decimal.) (3 points) 2. Fruit Fly Lifetimes: Suppose that the probability function shown in the table reflects the possible life-times (in months after emergence) for fruit trees. Fruit fly lifetimes (in months) x 1 2 3 4 5 6 p(x) 0.05 0.15 0.15 0.10 0.05 d. How long is a pregnancy if 90% of the pregnancies are shorter than this one? (A labeled sketch of the a. What proportion of fruit flies die in their first month? (2 pts) distribution, corresponding shaded areas, and documented calculator commands are required for full credit. Box final answer and round to three decimal places past the decimal.) (3 points) b. What is the probability that a fruit fly lives more than four months? (2 pts) c. What is the mean lifetime for a fruit fly? (3 pts)