Question 1208992: If P(A and B) = 0.2 and P (B|A) = 0.5, find P (A')
Found 2 solutions by math_tutor2020, greenestamps:
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
P(B | A) = P(A and B)/P(A)
P(A)*P(B|A) = P(A and B)
P(A)*0.5 = 0.2
P(A) = 0.2/0.5
P(A) = 2/5
P(A) = 0.4
P(A') = 1 - P(A)
P(A') = 1 - 0.4
P(A') = 0.6
Answer by greenestamps(13195)  (Show Source):
You can put this solution on YOUR website!
The solution from the other tutor is fine, using formal set notation.
An explanation in words is helpful to many students.
(1) P (B|A) = 0.5 means that B is true half of the time that A is true.
(2) P(A and B) = 0.2 means that A and B are both true 0.2 of the time.
The logical conclusion from (1) and (2) is that A is true but B is not also 0.2 of the time.
(3) So A is true and B is also true 0.2 of the time, and A is true and B is not 0.2 of the time, so A is true 0.2+0.2 = 0.4 of the time.
(4) And therefore A is not true 1-0.4 = 0.6 of the time.
ANSWER: P (A') = 0.6
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An alternative (and shorter) path, since different explanations work better for different students....
(1) P (B|A) = 0.5 means that B is true half of the time that A is true.
(2) P(A and B) = 0.2 means that A and B are both true 0.2 of the time.
(3) Since A and B are both true 0.2 of the time, and since B is true half the time that A is true, A is true twice as often as both A and B are true -- i.e., A is true 2(0.2) = 0.4 of the time.
(4) So A is not true 1-0.4 = 0.6 of the time.
ANSWER (again, of course): P (A') = 0.6
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