Let A and B be two events such that P(A) = 0.37 and P(B) = 0.03. Do not round your responses. (If necessary, consult a list of formulas.) (a) Determine P (AUB), given that A and B are mutually exclusive. (b) Determine P (A U B), given that A and B are independent.

Eastern Hemlock Ring shake, which is the separation of the wood between growth rings, is a serious problem in hemlock trees. Researchers have developed the following function that estimates the probability P that a given hemlock tree has ring shake: P(A,B,D)=11+e3.680.016A0.77B0.12D where A is the age of the tree yr, B is 1 if bird pecking is present and 0 otherwise, and D is the diameter in. of the tree at breast height. Source: Forest Products Journal. a. Estimate the probability that a 150-year-old tree with bird pecking present and a breast height diameter of 20in., will have ring shake. b. Estimate the probability that a 150-year-old tree, with no presence of bird pecking and a breast height diameter of 20in., will have ring shake. c. Develop a statement about what can be said about the influence have on the probability of ring shake. d. Using the total differential, estimate the probability if the actual age of the tree was 160 years and the diameter at breast height was 25in., Assume that no bird pecking was present. Compare your answer to the actual value. Hint: Assume that B=0 and exclude that variable from your calculations e. Comment on the practicality of using differentials in part d.

For the following exercises, use this scenario: a bag of M&Ms contains 12 blue, 6 brown, 10 orange, 8 yellow, 8 red, and 4 green M&Ms. Reaching into the bag, a person grabs 5 M&Ms. 50. What is the probability of getting no brown M&Ms?