~Conditional Probability~ rate(A|B) and rate(not A|B)

Question

Example if:

rate(A | B) = 10/180 x 100% ≈ 6%, rate(not A | B) = 170/180 x 100% ≈ 94%

Since rate(A | B) < rate(not A | B), is there association between A and B just based on the statements above?

I understand that if rate(A | B) < rate(A |not B), then there is a negative association with A and B. What about rate(A | B) < rate(not A | B)?

I appreciate any help. Thanks!

Answer 1

Let's suppose $\Pr(A | B) < \Pr(\neg A | B)$. Then, using Baye's theorem gives $$\Pr(B|A)\cdot\Pr(A)<\Pr(B|\neg A)\cdot\Pr(\neg A).$$ So we can see that the inequality might hold regardless of the relationship between $B$ and $A$. For example, if $A$ and $B$ are independent and $\Pr(A)<\Pr(\neg A)$, then it will hold. But it will also hold if $A$ and $B$ are negatively correlated and $\Pr(A)\leq\Pr(\neg A)$, as well as if $A$ and $B$ are positively correlated and $\Pr(A)\ll\Pr(\neg A)$, and other cases, too.

Answer 2

it is always true that $$P(A|B)+P(\bar A|B)=1 $$ So you only really have one piece of information.

You need more information, $P(A)$ would be nice.

You can conclude that the events are independent if $P(A|B) = P(A)$