(Applying Two-Way Frequency Tables LC)P(A) = 0.27, P(B) = 0.40, and P(A∩B) = 0.670. Are events A and B independent? Justify the answer mathematically.

(Interpreting Independence HC)The table summarizes the daily caffeine habits and majors of students at one university. Coffee Energy Drink No Caffeine STEM Major 0.32 0.13 0.02 Not STEM Major 0.17 0.27 0.09 Part A: Determine P(no caffeine | STEM major) and describe the event in everyday language. Show all work. (5 points)Part B: Are the events consuming no caffeine and a STEM major approximately independent? Use probabilities to justify the answer. (5 points)

(Addition and Multiplication Rules for Probability LC)P(A) = 0.53, P(B) = 0.21, and P(B | A) = 0.53. Are the events independent? Justify mathematically.

(Addition and Multiplication Rules for Probability LC)P(A) = 0.33, P(B) = 0.47, and P(A ∪ B) = 0.80. Are events A and B mutually exclusive? Justify mathematically.

(Combinations and Permutations LC)Calculate the number of combinations, given n = 6 and r = 3.

(Combinations and Permutations MC)A spelling bee has 22 contestants. In how many ways can they finish in first, second, and third place?

(Addition and Multiplication Rules for Probability LC)P(A) = 0.61, P(B) = 0.15, and P(A ∪ B) = 0.76. Are events A and B mutually exclusive? Justify mathematically.

(Set Operations and Relations MC)Given U = {g, h, i, j, k, l, m, n, o, p}, A = {g, h, m, o, p}, and B = {h, j, k, m, n, p}. If A′ ∩ B′ = {i, l}, determine (A ∪ B)′.

  (Logic Operations and Equivalence MC)Use the statements and truth table to determine which statements are equivalent. p: It is a university. q: It offers a degree in aviation management. p q ∼p ∼q p → q ∼p → ∼q q → p ∼q → ∼p T T F F T T T T T F F T F T T F F T T F T F F T F F T T T T T T

Hematopoiesis occurs in the _______ bone marrow.