Final answer:
To find the probability of events A and B occurring together (P(A ∩ B)), use the formula P(A ∩ B) = P(A) + P(B) - P(A ∪ B), substituting P(A) = x, P(B) = y, and P(A ∪ B) = z, which yields P(A ∩ B) = z - x - y. Independence or mutual exclusivity of events A and B can be determined based on the value of P(A ∩ B).
Step-by-step explanation:
Probability of Combined Events
The probability of events A and B occurring together, denoted as P(A ∩ B), can be found using the formula for the probability of the union of two events:
P(A ∩ B) = P(A) + P(B) - P(A ∪ B)
Given that P(A) = x, P(B) = y, and P(A ∪ B) = z, we can calculate P(A ∩ B) by rearranging the formula:
P(A ∩ B) = z - x - y
To determine if events A and B are independent or mutually exclusive, we look at the calculated probability of their intersection. If P(A ∩ B) equals the product of P(A) and P(B), the events are independent. However, if P(A ∩ B) is zero, the events are mutually exclusive.