probability

[shikh]

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 8 times the probability that both events A and B occur. What is the probability that event A occurs?
A.1/12
B.1/8
C.1/6
D.1/3
E.2/3
D
Source: kaplan test prep

[shankar.ashwin]

WHats the source of this problem? Question seems ambiguous to me.

Consider P(A) = 1/3 (From the answer choices)

WE know, P(A) = 2P(B)

Therefore P(B) = 1/6

Since 2 events are independent, their probability of occurring together is their product.

So P(A and B) = 1/3 * 1/6 = 1/18.

So in 18 trials, you have 1 outcome which has A and B occurring together.

In 18 trials, P(A) = 18/3 = 6

and, P(B) = 18/6 = 3.

So, P(A or B) = 6+3 = 9.

But then the questions mentions, ratio of

P(A and B) = 8* P(A or B)

I am guessing, we subtract the case of getting 'A and B' together from P(A or B) to get 8, but the questions states The probability that at least one of events A and B occurs is 8 times the probability that both events A and B occur

So I am guessing that would include getting A and B together as well, so ratio should be 1:9 and not 1:8. Pls correct me if I am wrong here.