Question 1: A bag contains 3 red, 2 blue, and 1 green ball. Two balls are drawn at random with replacement. Draw and label a tree diagram to show all possible outcomes and their probabilities.

Answer: See tree diagram with probabilities calculated as 1/6 for each draw, branching for each colour.

Question 2: Calculate the probability that both balls drawn are blue.

Answer: 1/36

Question 3: A card is drawn from a well-shuffled deck, then a coin is flipped. Construct a tree diagram for all outcomes and their probabilities.

Answer: Tree diagram with 52/52 and 1/2 branches, with combined probabilities.

Question 4: What is the probability of drawing a face card (Jack, Queen, King) and then flipping a head?

Answer: 3/13 * 1/2 = 3/26

Question 5: A three-stage tree shows the probability that a student passes each of three independent tests: P(pass) = 0.8, P(fail) = 0.2. Draw the tree and find the probability the student passes all three tests.

Answer: Probability = (0.8)^3 = 0.512

Question 6: In a complex tree, the probability of event A is 0.5, event B after A is 0.4, and event C after B is 0.3. Calculate the probability of the sequence A then B then C.

Answer: 0.5 * 0.4 * 0.3 = 0.06

Question 7: A complex tree diagram shows three independent events: Weather (Sunny or Rainy), Transport (Car or Bus), and Destination (School or Work). If the probability of sun is 0.6, car is 0.7, and going to school is 0.5, what is the probability that it is sunny, and the student goes to school by car?

Answer: 0.6 * 0.7 * 0.5 = 0.21

Question 8: Construct a tree diagram representing the scenario: A coin is tossed twice; record the probabilities of getting at least one head.

Answer: Tree with outcomes HH, HT, TH, TT; probability at least one head = 1 - P(TT) = 1 - 1/4 = 3/4.

Question 9: A box contains 4 defective and 16 non-defective items. Two items are chosen at random without replacement. Draw the tree diagram and find the probability that both items are non-defective.

Answer: (16/20) * (15/19) = 240/380 = 12/19

Question 10: Explain a common mistake when calculating probabilities in complex trees with independent events and suggest how to avoid it.

Answer: Common mistake: Multiplying probabilities incorrectly or assuming dependence. To avoid this, ensure events are independent and probabilities are multiplied correctly.

Question 11: In a complex tree, a student claims the probability of sequence A then B is 0.2, but the individual probabilities are P(A)=0.4 and P(B after A)=0.3. Is this claim correct? Explain.

Answer: No, because P(A then B) = 0.4 * 0.3 = 0.12, not 0.2. The claim is incorrect.