Question 1: A bag contains 3 red, 2 blue, and 4 green balls. A ball is drawn, then replaced. The process is repeated twice. Construct a tree diagram to represent all possible outcomes and their probabilities.

Answer: Tree diagram with probabilities calculated at each branch

Question 2: Calculate the probability that exactly two green balls are drawn in three draws, with replacement.

Answer: 0.312

Question 3: A fair six-sided die is rolled three times. Use a tree diagram to find the probability that the first roll is even, the second roll is a 3, and the third roll is greater than 4.

Answer: 1/2 * 1/6 * 2/6 = 1/36

Question 4: A machine produces three types of products: A, B, and C. The probabilities are 0.5, 0.3, and 0.2 respectively. Product A has two possible defects: minor and major. The probabilities of minor defect is 0.1, major defect is 0.05. Construct a tree diagram for this scenario and find the probability that a randomly selected product is either B or has a major defect.

Answer: Probability of B + Probability of A with major defect = 0.3 + (0.5 * 0.05) = 0.3 + 0.025 = 0.325

Question 5: In a three-stage experiment, the probabilities at each stage are independent: P(A)=0.6, P(B)=0.5, P(C)=0.4. Construct the tree diagram and determine the probability that all three events occur.

Answer: 0.6 * 0.5 * 0.4 = 0.12

Question 6: A student flips two coins and then rolls a die. Construct a tree diagram and find the probability that the student gets at least one head and rolls a 4 or 5.

Answer: Probability = P(at least one head) * P(roll 4 or 5) = (3/4) * (2/6) = 1/2

Question 7: Identify and correct the mistake in the following probability calculation based on a tree diagram: 'Probability of drawing a red ball then a blue ball (without replacement) is 3/9 * 2/8 = 6/72 = 1/12'.

Answer: Correct calculation: 3/9 * 2/8 = 6/72 = 1/12. However, if replacement is not specified, the probabilities change; without replacement, probabilities must be updated after the first draw.

Question 8: Construct a tree diagram for drawing two cards from a deck without replacement, and find the probability that both are kings.

Answer: 4/52 * 3/51 = 12/2652 = 1/221

Question 9: A box contains 5 defective and 15 non-defective items. Two items are randomly selected without replacement. Use a tree diagram to determine the probability that both items are defective.

Answer: 5/20 * 4/19 = 20/380 = 1/19

Question 10: A complex tree diagram shows three stages: first, choosing a color (Red, Blue, Green); second, selecting a shape (Circle, Square); third, selecting a size (Small, Large). If each choice is independent and has equal probability, what is the probability of selecting a Green, Square, Large combination?

Answer: 1/8