Question 1: A bag contains 3 red and 2 blue balls. Two balls are drawn at random with replacement. What is the probability that both balls are red? Show your working.

Answer: 0.36

Question 2: Construct a tree diagram for the following scenario: A coin is flipped twice. The first flip results in Heads or Tails with equal probability. The second flip depends on the first (if the first is Heads, the second is Heads with probability 0.6; if Tails, then Heads with probability 0.4).

Answer: Tree diagram with branches: H→H(0.6), H→T(0.4); T→H(0.4), T→T(0.6)

Question 3: A student answers incorrectly by multiplying marginal probabilities directly without considering the conditional probabilities. Identify and correct the mistake in this approach.

Answer: They need to multiply along the branches, considering conditional probabilities, not just marginal probabilities.

Question 4: Calculate the probability of drawing a red ball from Bag A (contains 4 red and 6 green) and then a green ball from Bag B (contains 5 red and 5 green), assuming independent draws.

Answer: 0.4 × 0.5 = 0.2

Question 5: In a tree diagram, the probability of event A is 0.3 and event B is 0.5. If these events are independent, what is the probability that both occur? Show your working.

Answer: 0.3 × 0.5 = 0.15

Question 6: A factory produces two types of products, X and Y. The probability of selecting X is 0.6, and Y is 0.4. If the probability that product X passes quality control is 0.8 and for Y is 0.7, what is the probability that a randomly chosen product is X and passes QC? Show your working.

Answer: 0.6 × 0.8 = 0.48

Question 7: Explain why multiplying marginal probabilities alone can lead to incorrect results when calculating joint probabilities in a tree diagram.

Answer: Because the events may not be independent; we need to consider conditional probabilities along the branches.

Question 8: A person flips a coin twice. If the first flip is Heads, the probability the second is Heads is 0.7; if Tails, then the probability the second is Heads is 0.3. Construct the corresponding tree diagram and find the probability that the person gets exactly one Heads.

Answer: Tree with branches: H→H(0.7), H→T(0.3); T→H(0.3), T→T(0.7). Probability of exactly one H: (H→T) + (T→H) = 0.3 + 0.3 = 0.6

Question 9: A survey shows that 20% of students prefer coffee and 30% prefer tea. The probability that a student who prefers coffee also likes tea is 0.5. Use a tree diagram to find the probability that a student prefers coffee and likes tea.

Answer: 0.2 × 0.5 = 0.1

Question 10: Identify and explain the common mistake made when students assume independence between events without verifying the information from the diagram.

Answer: Assuming independence when events may be dependent; need to check if the probabilities are conditional or marginal.

Question 11: A box contains 2 defective and 8 non-defective items. Two items are selected without replacement. Use a tree diagram to find the probability that both are non-defective.

Answer: P1 = 8/10, P2 = 7/9; total = (8/10) × (7/9) = 56/90 = 28/45