Final answer:
Event A and Event C are independent as the probability of both occurring together matches the product of their individual probabilities. We determine this by comparing P(A and C) to P(A) × P(C).
Step-by-step explanation:
To answer the student's question, we first need to understand the basics of probability and independence. Here are the steps to analyze the given events A, B, C with respect to independence:
- Event A = {1,3,5} is rolling an odd number.
- Event B = {2,4,6} is rolling an even number.
- Event C = {1,2,3,4} is rolling a number less than 5.
Two events are independent if the occurrence of one does not affect the probability of the occurrence of the other. The probability of event A, B, or C happening is 1/2, 1/2, and 2/3 respectively.
To check for independence, we calculate the probability of both events happening and compare this to the product of the probabilities of each event occurring independently. For example, P(A and B) = P(A) × P(B) if A and B are independent.
Now let's compute:
- P(A and B) = 0 (since there are no common outcomes).
- P(A and C) = P({1,3}) = 2/6 = 1/3.
- P(B and C) = P({2,4}) = 2/6 = 1/3.
P(A) × P(C) = (1/2) × (2/3) = 1/3 which matches P(A and C) and tells us that A and C are independent.
Event A is independent of Event C.
Similarly, we can check other combinations to identify the correct pairs of independent events.