The events A and B are independent, the probability that event A occurs is greater than 0, and the probability that event A occurs is twice the probability that event B occurs. The probability that at least one of events A and B occurs is 14 times the probability that both events A and B occur. What is the probability that event A occurs?  Choose the correct choice from the following options
A) 77/2 B) 1/7 C) 78/85 D) 1/5 E) 47/85
Let us say probability of A occuring is a.   Let us say probability of B occuring is b.   a = 2b   Probability (either A or B or both) =14 times Probability (A and B)   a*(1-b) + b*(1-a) + ab = 14*ab   Substituting a=2b in the second equation:   2b*(1-b) + b*(1-2b) + 2b*b = 14*2b*b   3b-2b^2 = 28b^2   3b = 30b^2   b = 3/30 = 1/10   So, a = 2b = 1/5   THE CORRECT ANSWER IS 1/5.
<result>D</result>
The events A and B are independent, the probability that event A occurs is greater than 0, and the probability that event A occurs is twice the probability that event B occurs. The probability that at least one of events A and B occurs is 14 times the probability that both events A and B occur. What is the probability that event A occurs?