Suppose you roll two standard number cubes. Let event A be rolling a 4 on the first die and let event B be rolling a 3 on the second die. Are A and B independent events? Are A and B mutually exclusive events? Explain.
(4 points)
Event A and event B are independent events. The outcome of rolling the first die does not affect the outcome of rolling the second die. The probability of rolling a 4 on the first die is 1/6, and the probability of rolling a 3 on the second die is also 1/6. The probability of both events A and B occurring is:
P(A and B) = P(A) * P(B) = 1/6 * 1/6 = 1/36
Since the probability of both events occurring is equal to the product of their individual probabilities, event A and event B are independent.
On the other hand, event A and event B are mutually exclusive events. Mutually exclusive events cannot occur at the same time. Rolling a 4 on the first die means that it is not possible to roll a 3 on the second die at the same time. Therefore, event A and event B cannot occur concurrently, making them mutually exclusive.