Final answer:
Event A (the sum of two dice is 7) and Event B (the second die shows an even number) are independent events. This is because the occurrence of B does not affect the probability of A occurring, and the probabilities P(A and B) and P(A)P(B) are equal.
Step-by-step explanation:
When considering whether two events are independent, we need to determine if the occurrence of one event affects the probability of the other event. Using an example of two six-sided dice, let's examine Event A (the sum of the dice is 7) and Event B (the second die shows an even number) to see if they are independent.
To find out if events A and B are independent, we can check if the probability of A occurring is the same regardless of whether B has occurred or not. The probability that the second die shows an even number (event B) is 1/2, since there are three even numbers (2, 4, 6) out of six possible outcomes on a die. Event A (the sum is 7) can happen in the following combinations: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). Since there are 36 total combinations when rolling two six-sided dice, the probability of A is 6/36 or 1/6.
To be independent, we would need P(A and B) to equal P(A)P(B). There are three combinations where the sum is 7 and the second die is even: (1,6), (5,2), and (3,4). Thus, P(A and B) = 3/36 = 1/12. Now we check if P(A)P(B) equals P(A and B): (1/6) * (1/2) = 1/12, which indeed equals 1/12. Since the probabilities are equal, events A and B are indeed independent.
In conclusion, the events A and B are independent events because the occurrence of one does not change the probability of the other, which is numerically justified by the equality of P(A and B) and P(A)P(B).