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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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    $P(B)$ 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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No lower bound: We give an example to show that the expression $\frac{P(B|A)-P(B)}{1-P(B)}$ can be arbitrarily large negative.

Imagine tossing a very unfair coin which has probability of head equal to $10^{-6}$, and therefore probability of tail equal to $1-10^{-6}$.

Let $B$ be the event "tail" and let $A$ the event "head." It is clear that $P(B|A)=0$. Thus our expression is equal to $\frac{0 -(1-10^{-6})}{1-(1-10^{-6})}.$ This simplifies to $-999999$.

By choosing $10^{-66}$ instead of $10^{-6}$, we can make our expression inconceivably huge negative. So there is no universal lower bound for our expression. (If suitable restrictions are put on $B$ and $A$, there may be a lower bound.)

An upper bound: Choose any $A$ such that $P(B|A)=1$, for example choose $A=B$. Then our expression is equal to $1$. It cannot ever be larger than $1$, since for given $P(B)$, the numerator is maximized if $A$ is such that $P(B|A)=1$. Thus $1$ is an upper bound for our expression. This upper bound can be attained, so there is no cheaper universal upper bound.

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    Your meaning is not clear to me. One can make sense of $A \implies B$ in a probabilistic context, but you would have to frame a specific question.2011-10-04
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As another way of looking at André Nicolas's solution, note that your expression is the same as $1-\dfrac{\Pr(\text{not }B|A)}{\Pr(\text{not }B)}$ where the quotient can take any non-negative value and so the whole expression can take any value from 1 downwards including any negative value.

As he says in a comment, this may be a function of probabilities but it is not a probability itself.