Question

Given a sample space (S = a, b, c, d, e, f, g) and events (A = a, b, c, d) and (B = f, g), with probabilities known, determine:

a) (P(A cup B))
b) (P(A' | B))

Answer 1

The probability of event A or B occurring in the given sample space is 1 or 100%, and the conditional probability of the complement of A given B is equal to the probability of B.

Step-by-step explanation:

The question involves calculating combined probabilities and conditional probabilities within a sample space. To calculate P(A ∪ B), which is the probability of event A or event B occurring, we add the probabilities of A and B, provided they are mutually exclusive, which they are in this case as they have no common outcomes. Given the provided sample space S = {a, b, c, d, e, f, g} and events A = {a, b, c, d} and B = {f, g}, the union of A and B is equal to the entire sample set S since all elements are covered in A and B, therefore P(A ∪ B) is 1 or 100%.

Next, we look at the conditional probability of the complement of A given B, denoted as P(A' | B). This is basically asking what the probability of not being in event A is, given that we are within the outcomes of event B. Since B = {f, g} and A' (the complement of A) contains {e, f, g}, the overlap between A' and B is actually events B themselves. Thus, P(A' | B) = P(B) because the entirety of B's outcomes are within A'.