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How do you find the range ( min and max) for a probability function such as $$\frac{P (B|A) − P (B)} {1−P (B)}\;?$$

What I tried was to use Venn diagrams, but I couldn't find a solution as the circles must overlap completely to max P (B|A). But that causes P(B) to decrease.

Any thoughts?

Thank you.

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    Presumably you have some concrete problem in mind. Without further information, $P(B|A)$ could be $0$, or it could be $1$, or anywhere in between. The ratio you wrote down could be huge negative. It is never greater than $1$. If there are additional details that you know, perhaps a more explicit answer could be supplied.2011-10-04
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    That is all to the question. the value of P(B|A) can be anything. So the upper bound of the equation is 1. How would one solve to a possible lower bound. Thank you.2011-10-04
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    I will give an answer to the lower bound issue, because it is unpleasant to type it as a series of comments.2011-10-04
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    When does the probability function get its minimum and maximum values.2011-10-04
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    I am concerned about your use of the term "probability function." For a probability function, you would need first of all a sample space. And as has been observed in my comments and answer, the expression you gave can in some situations be negative. Then certainly it cannot be interpreted as a probability.2011-10-04
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    @Jimmy: $P(B|A)=\frac{P(B\cap A)}{P(A)}$, so $P(B|A)$ can range from $0$ to $1$. So a lower limit to the expression would be $\frac{-P(B)}{1-P(B)}$, which is negative, as André commented.2011-10-04
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    $P(B)$ is bounded above by $\max\{P(B\mid A), P(B\mid A^c)\}$ and bounded below by $\min\{P(B\mid A), P(B\mid A^c\}$.2011-10-04

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