Question 1: A bag contains 3 red, 2 blue, and 4 green balls. A ball is drawn at random, then replaced. The process is repeated twice. Construct a tree diagram to represent all possible outcomes, and calculate the probability of drawing two red balls in succession.

Answer: 1/9

Question 2: A die is rolled three times. For each roll, the outcome is independent. Using a tree diagram, find the probability that exactly two of the rolls show a 6.

Answer: 1/16

Question 3: A factory produces three types of smartphones: A, B, and C. The probability of a defect is 0.02 for A, 0.03 for B, and 0.01 for C. A batch of 300 phones is produced with proportions 50% A, 30% B, and 20% C. Construct a tree diagram to show the probability that a randomly selected phone is defective, including the proportions from each type.

Answer: Approximately 0.017

Question 4: A student answers 4 multiple-choice questions, each with 4 options, where one option is correct. Assuming independence, construct a tree diagram and calculate the probability the student answers exactly 2 questions correctly.

Answer: 0.264

Question 5: In a game, a player rolls a die and flips a coin simultaneously. Construct a combined tree diagram for the two events, and determine the probability that the player rolls a 4 and the coin shows heads.

Answer: 1/12

Question 6: A survey finds that 70% of students like chocolate ice cream, 50% like vanilla, and 40% like both. Construct a tree diagram to model these preferences, and find the probability that a student likes either chocolate or vanilla.

Answer: 0.8

Question 7: A traffic light can be green, yellow, or red. The probability of green is 0.6, yellow 0.3, and red 0.1. A car passes the light twice independently. Construct a tree diagram and find the probability that the car passes at least once while the light is green.

Answer: 0.84

Question 8: An online store ships orders in three regions: North, South, and East. The probability of a delay is 0.05 in North, 0.10 in South, and 0.02 in East. Two orders are sent independently. Construct a tree diagram and find the probability that at least one order is delayed.

Answer: 0.196

Question 9: Two independent events—flipping a coin and rolling a die—occur sequentially. Construct a tree diagram representing all outcomes and explain why the probability of both occurring in a specific way (e.g., Heads and rolling a 3) is the product of their individual probabilities.

Answer: Because the events are independent, multiply their probabilities: 1/2 * 1/6 = 1/12.

Question 10: Identify a common mistake when constructing a tree diagram for three independent events and explain how to correct it.

Answer: Incorrectly linking branches or not multiplying probabilities at each stage. Ensure each branch multiplies the prior probability by the stage's event probability.

Question 11: Challenge Question: A chocolate box contains 5 milk, 3 dark, and 2 white chocolates. Two chocolates are selected at random without replacement. Construct a tree diagram and find the probability that the first is white and the second is milk.

Answer: 1/15