You have a fair ?ve-sided die. The sides of the die are numbered from 1 to 5. Each die roll is independent of all others, and all faces are equally likely to come out on top when the die is rolled. Suppose you roll the die twice.
(a) Let event A to be "the total of two rolls is 10", event B be "at least one roll resulted in 5", and event C be "at least one roll resulted in 1".
i. Is event A independent of event B?
ii. Is event A independent of event C?
(b) Let event D be "the total of two rolls is 7", event E be "the di?erence between the two roll outcomes is exactly 1", and event F be "the second roll resulted in a higher number than the ?rst roll".
i. Are events E and F independent?
ii. Are events E and F independent given event D?