Question 1128416
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P(A) = 0.3
P(B) = 0.8
P(A and B) = 0.24


P(A | B) = P(A and B)/P(B)
P(A | B) = 0.24/0.8
P(A | B) = 0.3
P(A | B) = P(A) <--- we'll use this fact


P(B | A) = P(B and A)/P(A)
P(B | A) = P(A and B)/P(A)
P(B | A) = 0.24/0.3
P(B | A) = 0.8
P(B | A) = P(B) <---- and this fact too


Since both <font color=red>P(A | B) = P(A) and P(B | A) = P(B)</font> are true, this means that <font color=red>A and B are independent events</font>.


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Put another way: if we know that event A occurs, then that does not change the probability P(B). So P(B | A) = P(B) meaning B is independent of A. Similarly, P(A | B) = P(A) tells us that prior knowledge of event B happening does not change the probability P(A), therefore A is independent of B. 


If either P(A|B) = P(A) or P(B|A) = P(B) were false equations, then A and B would be dependent events. 
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