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# The events A and B are independent, the probability that

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The events A and B are independent, the probability that [#permalink]  24 Mar 2020, 08:58
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58% (01:47) correct 41% (03:27) wrong based on 12 sessions
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
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Re: The events A and B are independent, the probability that [#permalink]  24 Mar 2020, 09:39
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Expert's post
a = 2b

Probability (either A or B or both) = 8 times Probability (A and B)
a*(1-b) + b*(1-a) + ab = 8*ab

Substituting a=2b in the second equation:

2b*(1-b) + b*(1-2b) + 2b*b = 8*2b*b

3b-2b^2 = 16b^2
3b = 18b^2
b = 3/18 = 1/6

So, a = 2b = 1/3
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Intern
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Re: The events A and B are independent, the probability that [#permalink]  25 Mar 2020, 05:44
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P(A or B) = P(A) + P(B) - P(A and B).....(1)
since A and B are independent, then P(A and B)= P(A) * P(B)
Given,
P(A or B) = 8*P(A and B)
P(A) = 2*P(B) ; P(B) = P(A)/2
therefore,
8* P(A and B) = 8 (P(A) *P(A)/2) = 4 (P(A)^2)
substituting the given values into (1),
4(P(A)^2) = P(A) + P(A)/2 -P(A)^2)/2
4(P(A)^2) + P(A)^2/2 =3P(A)/2
9(P(A)^2)/2 = 3P(A)/2
Hence,
P(A) =1/3
Re: The events A and B are independent, the probability that   [#permalink] 25 Mar 2020, 05:44
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