SOLUTION: If P(A|B) = 1, must A = B? Use a Venn diagram to explain your answer

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Question 1121847: If P(A|B) = 1, must A = B? Use a Venn diagram to explain your answer
Found 2 solutions by solver91311, ikleyn:
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


No.

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John

My calculator said it, I believe it, that settles it

Answer by ikleyn(52898) About Me  (Show Source):
You can put this solution on YOUR website!
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DEFINITION : 

    P(A|B) ≡ the (conditional) Probability of A given B occurs


If A and B are infinite sets, the answer is NOT NECESSARY positive.

Sometime it can be negative.

You will find several examples below. 

1.   Let B = the set of all integer numbers and A = the subset of integer numbers with the absolute values greater than 5   ( |n| > 5 ).


     Then P(A|B) = 1, but A =/= B.


     In the "probability" terms, the probability that randomly selected integer number belongs to A is equal to  1,  but  A =/= B.




2.   Let B = the set of all real numbers and A = the set of all real numbers except of x= 0   (the real number line with the hole at x= 0).


     Then P(A|B) = 1, but A =/= B.

 
     In the "probability" terms, the probability that randomly selected point from B belongs to A is equal to  1,  but  A =/= B.




3.   Let B = the set of all complex numbers (a complex plane) and A be the subset of all non-real numbers.


     Then P(A|B) = 1,  but  A =/= B.


     In the "probability" terms, the probability that randomly selected point from B belongs to A is equal to  1,  but  A =/= B.  



4.   Let B = the set of all points inside and at the border of the circle of the radius of "r", and A is the subset of all interior points.


     Then P(A|B) = 1,  but  A =/= B.


     In the "probability" terms, the probability that randomly selected point from B belongs to A is equal to  1,  but  A =/= B.  



5.   Let B = the set of all whole integer numbers, and A is the subset of all composite numbers.


     Then P(A|B) = 1,  but  A =/= B.


     In the "probability" terms, the probability that randomly selected whole number is a composite number is equal to  1,  but  A =/= B.