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The original problem, as it is posted, worded and printed, creates many questions to its content.
To avoid these questions, I will re-formulate the problem in THIS FORM :
There is a universal finite set U and two its subsets A and B.
The experiment is to select randomly some element from the universal set U.
The probability to select randomly any element from U is the same for all elements of U.
With probability 0.3, the selected element is from subset A; with probability 0.2, the selected element is from subset B.
The selection is independent between subsets A and B.
If an element is selected from the union (A U B), what is the probability that it is from the intersection (A ∩ B) ?
SOLUTION
The problem asks about the conditional probability P = P (A ∩ B) / P(A U B).
Due to the independence, P(A ∩ B) = P(A)*P(B) = 0.3*0.2 = 0.06.
From the general Probability Theory formula, P(A U B) = P(A) + P(B) - P(A ∩ B) = 0.3 + 0.2 - 0.06 = 0.44.
THEREFORE, the final conditional probability under the problem's question is P = = 0.136364 (rounded). ANSWER
Solved.