Question 1: A survey shows that 60% of students like football, 40% like basketball, and 20% like both. Calculate the probability that a randomly selected student likes either football or basketball, but not both.

Answer: 0.6 + 0.4 - 2×0.2 = 0.6 + 0.4 - 0.4 = 0.6

Question 2: In a Venn diagram, if the probability of being in set A is 0.5, set B is 0.3, and the intersection is 0.1, what is the probability that a student is in A or B?

Answer: 0.5 + 0.3 - 0.1 = 0.7

Question 3: A bag contains 10 red balls, 15 blue balls, and 5 green balls. If a ball is chosen at random, what is the probability it is not green?

Answer: (10 + 15)/30 = 25/30 = 5/6

Question 4: A student says, 'The probability of selecting a red card from a deck is 1/4.' Is this correct? Explain your reasoning.

Answer: Yes, because there are 13 red cards in a 52-card deck, so 13/52 = 1/4.

Question 5: Construct a Venn diagram to represent the following: 70% like tennis, 50% like cricket, and 20% like both. Calculate the probability that a student likes neither sport.

Answer: Probability in either sport: 0.7 + 0.5 - 0.2 = 1.0 Probability in neither: 1 - 1.0 = 0

Question 6: A survey shows that 55% of people prefer tea, 35% prefer coffee, and 10% prefer both. What is the probability that a person prefers only tea?

Answer: 0.55 - 0.10 = 0.45

Question 7: A common mistake is to add the probabilities of two sets without subtracting the intersection. Explain why this is incorrect with an example.

Answer: Because adding probabilities without subtracting the intersection counts the overlap twice, leading to an overestimation. For example, if P(A)=0.4, P(B)=0.3, and P(A∩B)=0.1, adding 0.4 + 0.3 = 0.7 overcounts the shared part.

Question 8: If 25% of students like only sport A, 15% like only sport B, and 10% like both, what is the probability that a student does not like either sport?

Answer: 1 - (0.25 + 0.15 + 0.10) = 1 - 0.50 = 0.50

Question 9: A factory produces 5% defective items. If you select 20 items at random, what is the probability that exactly 1 item is defective? (Use the binomial probability formula.)

Answer: P(X=1) = C(20,1)×0.05×(0.95)^19 ≈ 20×0.05×(0.95)^19

Question 10: Identify and correct the mistake in the following reasoning: 'Since the probability of event A is 0.6 and event B is 0.5, the probability of either A or B should be 1.1.'

Answer: The probability cannot exceed 1. Adding 0.6 and 0.5 gives 1.1, but the correct calculation is 0.6 + 0.5 - P(A∩B), which is less than or equal to 1.