Final answer:
The probability of event A, given that A and B are independent and the probability that neither occurs is f, while the probability of B is g, is calculated as P(A) = (1 - f - g) / (1 - g).
Step-by-step explanation:
Given that A and B are independent events, the probability that neither occurs is denoted by f, and the probability of B occurring is denoted by g. We can use the properties of probabilities for independent events to find the probability of A (P(A)). Since A and B are independent, the probability that both A and B occur equals the product of their individual probabilities, meaning P(A AND B) = P(A)P(B). Furthermore, the probability that neither A nor B occurs (meaning both events do not happen) is equal to 1 minus the probability that at least one occurs, which means f = 1 - (P(A OR B)). For independent events, P(A OR B) = P(A) + P(B) - P(A AND B).
Combining these formulas, we can express f as:
f = 1 - (P(A) + g - P(A)g)
Solving for P(A) gives us:
P(A) = (1 - f - g)/(1 - g)