Two fair dice, one red and one blue, are rolled.

Let A be the event that the number rolled on the red die is odd.

Let B be the event that the number rolled on the blue die is odd.

Let C be the event that the sum of the numbers rolled on the two dice is odd. Determine which of the following is true.

• A, B, and C are not mutually independent, but each pair is independent.
• A, B, and C are mutually independent.
• Exactly one pair of the three events is independent.
• Exactly two of the three pairs are independent.
• No pair of the three events is independent.

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: A

Each event has probability 0.5. Each of the three possible intersections of two events has probability 0.25 = (0.5)(0.5), so each pair is independent. The intersection of all three events has probability 0, which does not equal (0.5)(0.5)(0.5) and so the three events are not mutually independent.

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.