Question 1: A bag contains 3 red balls and 2 blue balls. Two balls are drawn at random without replacement. Using a tree diagram, find the probability that both balls are red.

Answer: 3/5 * 2/4 = 3/10

Question 2: A coin is tossed twice. Construct a tree diagram showing all possible outcomes and their probabilities. What is the probability of getting exactly one head?

Answer: 1/2 * 1/2 + 1/2 * 1/2 = 1/4 + 1/4 = 1/2

Question 3: A factory produces two types of products, A and B. The probability that a product passes quality control is 0.9 for A and 0.8 for B. If a product is randomly selected and found to have passed, what is the probability it is type A?

Answer: P(A|pass) = (0.5*0.9) / (0.5*0.9 + 0.5*0.8) = 0.45 / (0.45 + 0.4) = 0.45/0.85 = 9/17

Question 4: In a two-step process, the probability of success at stage 1 is 0.6 and at stage 2 is 0.7, independent of each other. Use a tree diagram to find the probability that both stages succeed.

Answer: 0.6 * 0.7 = 0.42

Question 5: A student randomly guesses answers on a quiz with two multiple-choice questions. Each question has 4 options. Construct a tree diagram for all outcomes and find the probability the student gets exactly one question correct.

Answer: Correct on Q1 only: 1/4 * 3/4 = 3/16; Correct on Q2 only: 3/4 * 1/4 = 3/16; Total = 3/16 + 3/16 = 6/16 = 3/8

Question 6: Construct a tree diagram showing the probabilities of selecting a letter 'A' or 'B' from one box (probability 0.6 for A, 0.4 for B) and then a letter 'X' or 'Y' from another box (probability 0.7 for X, 0.3 for Y). Find the probability of selecting 'A' and then 'Y'.

Answer: 0.6 * 0.3 = 0.18

Question 7: A card is drawn from a deck twice with replacement. Construct a tree diagram and determine the probability that both cards drawn are hearts.

Answer: 13/52 * 13/52 = 1/16

Question 8: A machine produces parts with a 5% defect rate. Two parts are selected at random, independently. Use a tree diagram to find the probability that both parts are non-defective.

Answer: (1 - 0.05) * (1 - 0.05) = 0.95 * 0.95 = 0.9025

Question 9: A survey shows 70% of people like tea and 50% like coffee. If 20% like both, construct a tree diagram showing these probabilities. What is the probability a person likes only tea?

Answer: P(tea only) = 0.7 - 0.2 = 0.5

Question 10: A die is rolled twice. Construct a tree diagram and find the probability that the first roll is even and the second roll is odd.

Answer: (1/2) * (1/2) = 1/4

Question 11: Identify and correct the mistake in this probability calculation: 'Probability both events occur: 0.6 * 0.7 = 0.42'. The events are dependent with P(second success|first success) = 0.5.

Answer: Incorrect; should multiply 0.6 by 0.5, giving 0.6 * 0.5 = 0.3