Final answer:
The correct set notation for even counting numbers between 3 and 13 is not listed in the given options, but it would be {4, 6, 8, 10, 12}. Using set theory and probability with given sample spaces and events, you can also determine probabilities and intersections for various scenarios.
Step-by-step explanation:
The set notation for even counting numbers between 3 and 13 is x is an even counting number between 3 and 13. However, since the even numbers must also be greater than 3, the set includes 4, 6, 8, 10, and 12 (option b is incorrect because it includes numbers between 1 and 15, and option d is incorrect because it describes odd numbers). Therefore, the mentioned correct answer in the final answer is actually not in the provided options, but it would be {4, 6, 8, 10, 12}.
In terms of set notation and events, if the sample space S is the whole numbers starting at one and less than 20, then S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}. Let event A be the even numbers, A = {2, 4, 6, 8, 10, 12, 14, 16, 18}, and event B be the numbers greater than 13, B = {14, 15, 16, 17, 18, 19}.
The probability of A, P(A), is calculated by the number of outcomes in A divided by the number of outcomes in S, so P(A) = 9/19. The set A AND B contains the outcomes that are both even and greater than 13, so A AND B = {14, 16, 18}.