SOLUTION: Let A and B be events with P(A)=1/2, P(B)=1/3 and P(A∩B)=1/4. Find i) P(A | B) ii) P(B | A) iii) P(A U B) iv) P(A'| B')

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Question 1182451: Let A and B be events with P(A)=1/2, P(B)=1/3 and P(A∩B)=1/4. Find
i) P(A | B)
ii) P(B | A)
iii) P(A U B)
iv) P(A'| B')
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52803)   (Show Source): You can put this solution on YOUR website!
.
Let A and B be events with P(A)=1/2, P(B)=1/3 and P(A∩B)=1/4. Find
i) P(A | B)
ii) P(B | A)
iii) P(A U B)
iv) P(A'| B')
~~~~~~~~~~~~~~~~~~~~~ 

(i)    P(A|B) = P(A∩B)/P(B) =  = .


(ii)   P(B|A) = P(A∩B)/P(A) =  =  = .


(iii)  P(AUB) = P(A) + P(B) - P(A∩B) =  +  -  =  = .


(iv)  P(A'|B') = P(A'∩B')/P(B') = (1-P(AUB))/(1-P(B)) =  =  = = .

Solved.



Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


Use 12 as the denominator for each probability, since 12 is the least common multiple of 2, 3, and 4.

Given: P(A) = 6/12; P(B) = 4/12; P(A∩B) = 3/12

Then
P(A∩B') = P(A)-P(A∩B) = 6/12-3/12 = 3/12
P(A'∩B) = P(B)-P(A∩B) = 4/12-3/12 = 1/12

P(AUB) = P(A)+P(B)-P(A∩B) = 6/12+4/12-3/12 = 7/12
P(A'∩B') = 1-P(AUB) = 1-7/12 = 5/12

i) P(A|B) = P(A∩B)/P(A) = (3/12)/(6/12) = 3/6 = 1/2

ii) P(B|A) = P(A∩B)/P(B) = (3/12)/(4/12) = 3/4

iii) (from above) P(AUB) = 7/12

iv) P(A'|B') = P(A'∩B')/P(B') = (5/12)/(1-4/12) = (5/12)/(8/12) = 5/8
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