Final answer:
This mathematics question asks for the understanding and calculation of probabilities, including the intersection of events, independent events, mutually exclusive events, and conditional probabilities, typically at a high school level.
Step-by-step explanation:
The question presented involves using probability to understand different scenarios and outcomes, which is a typical problem in the field of mathematics. Specifically, it is focused on combining and comparing probabilities, and how to calculate the probability of the intersection of events (P(A AND B)), and also touches upon the concepts of independent events, mutually exclusive events, and conditional probabilities.
Understanding Probabilities through a Venn Diagram
Creating a Venn diagram is an excellent way to visualize the intersection and union of sets, as well as the probabilities of those sets occurring. Label each section of the Venn diagram to represent the respective probabilities of events A and B, as well as their intersection A ∩ B. This visual aid can help clarify the relationships between these events and the calculations that are based on them.
Rules of Probability
The multiplication rule and the addition rule are two fundamental principles used in probability. The multiplication rule states that the probability of both events A and B occurring (P(A AND B)) is equal to the probability of A given B is true (P(A|B)) times the probability of B (P(B)). Conversely, the addition rule states that the probability of either event A or B occurring (P(A OR B)) is the sum of the probabilities of A and B minus the intersection (P(A AND B)).
Independent and Mutually Exclusive Events
To determine if events are independent, you check if P(A AND B) is equal to P(A) multiplied by P(B). If the probabilities are not equal, the events are dependent. For events to be mutually exclusive, they cannot happen at the same time, and thus the intersection should be zero.