Question

Let (Ω, F , P) be a probability space. Suppose that Ω is the collection of all possible outcomes of a single iteration of a certain experiment. Also suppose that, for each C ∈ F, the probability that the outcome of this experiment is contained in C is P(C).
Consider events A, B ∈ F with P(A) + P(B) > 0. Suppose that the experiment is iterated indefinitely, with each iteration identical and independent of all the other iterations, until it results in an outcome that is an element of A ∪ B, after which it stops. What is the probability that this procedure results in an outcome that is an element of A?
Your answer does not have to include an explicit probability model, but it should be properly motivated. You should avoid using conditional probabilities.

Answer 1

it has been mentioned that procedure stops only when result in an out come that is an element of A U B , Hence The procedure results in an out come that is an element of A only happens during  last experiment .

Hence probability of procedure results in an outcome that is an element of A is the equal to probability of an outcome that is an element of A during last experiment

Given last experiment result in an outcome that is an element of A U B ,hence probability of procedure results in an outcome that is an element of A equal to [P(A (AB)]/P(AB) = P(A)/P(AB) as A(AB)=A