Final answer:
To find the probabilities of the events A, B, C, and D in Alice and Bob's random number selection, we consider the possible outcomes and calculate the probabilities. P(A) = 2/9, P(B) = 7/9, P(C) = 1/3, and P(D) = 2/3.
Step-by-step explanation:
To find the probabilities of the events A, B, C, and D, we need to understand the concepts of probability and independent events. In this case, Alice and Bob each choose a number between zero and two, so there are a total of 3 x 3 = 9 possible outcomes. Let's calculate the probabilities for each event:
A: The magnitude of the difference of the two numbers is greater than 1/3. There are two possible outcomes where this event occurs: (0, 2) and (2, 0). Therefore, P(A) = 2/9.
B: At least one of the numbers is greater than 1/3. There are seven possible outcomes where this event occurs: (1, 0), (2, 0), (0, 1), (0, 2), (1, 1), (1, 2), (2, 1). Therefore, P(B) = 7/9.
C: The two numbers are equal. There are three possible outcomes where this event occurs: (0, 0), (1, 1), (2, 2). Therefore, P(C) = 3/9 = 1/3.
D: Alice's number is greater than 1/3. There are six possible outcomes where this event occurs: (1, 0), (2, 0), (1, 1), (1, 2), (2, 1), (2, 2). Therefore, P(D) = 6/9 = 2/3.