Question 1: A bag contains 3 red, 2 blue, and 4 green balls. A ball is randomly drawn, replaced, and then a second ball is drawn. Construct a tree diagram for the possible outcomes of the two draws and find the probability that both balls drawn are blue.

Answer: Probability = (2/9) * (2/9) = 4/81

Question 2: Calculate the probability that, in three independent coin tosses, at least two heads appear. Construct the corresponding tree diagram to illustrate your reasoning.

Answer: P(at least 2 heads) = 3/8

Question 3: A survey shows that 60% of students like chocolate, 25% like vanilla, and 15% like both. Construct a tree diagram to represent this information and find the probability that a randomly selected student likes either chocolate or vanilla.

Answer: P(Chocolate or Vanilla) = 0.60 + 0.25 - (0.60*0.25) = 0.60 + 0.25 - 0.15 = 0.70

Question 4: Construct a tree diagram representing the probability of selecting a card from a deck (without replacement), where the card is a heart or not, followed by drawing a face card or not from the remaining cards. Calculate the probability that the first card is a heart and the second is a face card.

Answer: Probability = (13/52) * (3/51) = 1/17

Question 5: A farmer randomly selects 3 eggs from a batch of 10, of which 4 are defective. Construct a tree diagram to show the probability of selecting 0, 1, 2, or 3 defective eggs. Calculate the probability that exactly 2 defective eggs are chosen.

Answer: P(exactly 2 defective) = (4/10)*(3/9)*(6/8) + other arrangements = 0.308

Question 6: A box contains 5 red, 3 blue, and 2 yellow balls. Two balls are drawn sequentially without replacement. Construct a tree diagram to determine the probability that both balls are of different colours.

Answer: Probability = 1 - (both same colour) = (R,R)+(B,B)+(Y,Y) = ... = 0.66 approx.

Question 7: A machine produces parts, each with a 5% defect rate. If three parts are selected independently, construct the tree diagram and find the probability that at most one is defective.

Answer: Probability = P(0 defectives) + P(1 defective) = (0.95)^3 + 3*(0.05)*(0.95)^2 = 0.857 + 3*0.05*0.9025 = 0.857 + 0.135 = 0.992

Question 8: A student flips a coin, then rolls a die. Construct a tree diagram for the combined outcomes, and calculate the probability of getting a head and a 4.

Answer: Probability = (1/2)*(1/6)=1/12

Question 9: In a game, a spinner is divided into 4 equal sectors numbered 1 to 4. A fair die is rolled. Construct a tree diagram to represent possible outcomes. Find the probability that the spinner shows an even number and the die shows a number greater than 2.

Answer: Probability = (2/4)*(2/6)=1/6

Question 10: Identify and correct the mistake in the following probability calculation: 'Probability of A or B = 0.3 + 0.4 = 0.7', assuming A and B are independent events with probabilities 0.3 and 0.4 respectively.

Answer: Correct probability = P(A) + P(B) - P(A∩B) = 0.3 + 0.4 - (0.3*0.4) = 0.3 + 0.4 - 0.12 = 0.58