Final answer:
Upon evaluating the given sets of possible outcomes A, B, and C when a fair die is rolled, it is found that only events B and C are independent. Events A and B, and A and C are not independent because their combined probabilities do not equal the product of their individual probabilities in the case of A and C, and A and B are mutually exclusive with no overlapping outcomes.
Step-by-step explanation:
The question involves determining which pairs of events are independent events when a fair die is rolled once. Events A, B, and C are given as subsets of possible outcomes when rolling a fair die. To determine if two events are independent, we use the criterion that P(A AND B) should equal P(A)P(B), or equivalently, P(B|A) should equal P(B), meaning that the occurrence of event A does not affect the probability of event B occurring.
For events A and C to be independent, P(A AND C) should equal P(A)P(C). Event A consists of rolling an odd number (1, 3, or 5), and event C consists of rolling a number less than 5 (1, 2, 3, or 4). The intersection of A and C is {1, 3}, which occurs with probability 2/6 since there are two favorable outcomes out of six possible outcomes when rolling a die. P(A) for the event of rolling an odd number is 3/6 (since there are three odd numbers out of six possible outcomes). P(C) for the event of rolling a number less than 5 is 4/6 (since there are four numbers out of six that are less than 5). If A and C were independent, then P(A)P(C) should equal P(A AND C), which would be (3/6)*(4/6). However, 12/36 does not equal 2/6, so A and C are not independent.
For events B and C to be independent, similar logic applies. The intersection of B and C is {2, 4}, which occurs with probability 2/6. P(B) is 3/6 and P(C) is 4/6. (3/6)*(4/6) equals 12/36 or 2/6, which is equal to the probability of P(B AND C). Therefore, B and C are independent events.
Finally, for events A and B, since they have no outcomes in common (A is {1,3,5} and B is {2,4,6}), the intersection of A and B is the empty set, and thus P(A AND B) is 0. Because P(A)P(B) is not 0 (as both probabilities are non-zero), A and B are dependent events. They are actually mutually exclusive since there are no outcomes that satisfy both events at the same time.
Therefore, the correct answer is that B and C are independent only.