SOLUTION: Assume that P(A) = 0.7, P(B) = 0.8, and P(B and A) = 0.56. a) Find P(A|B) and P(B|A). b) Are events A and B independent? Explain.

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Question 1197190: Assume that P(A) = 0.7, P(B) = 0.8, and
P(B and A) = 0.56.
a) Find P(A|B) and P(B|A).
b) Are events A and B independent? Explain.
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52754)   (Show Source): You can put this solution on YOUR website!
.

(a)  P(A|B) =  =  = 0.7.    ANSWER


     P(B|A) =  =  = 0.8.    ANSWER



(b)  Events A and B are independent         ANSWER  

         because  P(A and B) = 0.56  is numerically the same as  P(A)*P(B) = 0.7*0.8 = 0.56.

Solved.



Answer by math_tutor2020(3816)   (Show Source): You can put this solution on YOUR website!

The vertical bar represents "given"
P(A | B) = P(A given B)
For more information, search out "conditional probability".

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Part (a)

P(A given B) = P(A and B)/P(B)
P(A given B) = 0.56/0.8
P(A given B) = 0.7
and
P(B given A) = P(B and A)/P(A)
P(B given A) = P(A and B)/P(A)
P(B given A) = 0.56/0.7
P(B given A) = 0.8

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Part (b)

There are two ways we can prove A and B are independent.

Method 1
Note how P(A given B) = 0.7 = P(A)
This shows that the prior event B occurring does NOT change the result of P(A)
Similarly P(B given A) = 0.8 = P(B) showing prior event A occurring does not change P(B)
Neither event affects the other; hence they are independent.

Method 2
If A and B were independent, then the product of P(A) and P(B) should get us P(A and B)
P(A and B) = P(A)*P(B)
P(A and B) = 0.7*0.8
P(A and B) = 0.56
This matches the P(A and B) value in the instructions
We've confirmed that P(A and B) = P(A)*P(B) which shows the events to be independent.
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