I know that in general $\Pr(A|B) = 1 - \Pr(A'|B)$. However, I'm not sure how this works for the following exmaple:
Consider some random number $x\in[0, \infty)$. Now consider the conditional probability: $$\Pr(x\in[z, z+\delta)|x\not\in[0, z))$$ for some $z\in[0, \infty)$ and $\delta>0$. Expanding this gives: $$\frac{\Pr(x\in[z, z+\delta) \cap x\not\in [0,z))}{\Pr(x\not\in[0,z))}$$
But since $x$ is a unique number, $x\in[z, z+\delta) \implies x\not\in[0,z)$. So this then becomes: $$\Pr(x\in[z, z+\delta)|x\not\in[0, z))=\frac{\Pr(x\in[z, z+\delta))}{\Pr(x\not\in[0, z))}$$
Now consider the 'complement' event. That is: $$\Pr(x\not\in[z, z+\delta)|x\not\in[0,z))$$
Expanding this gives: $$\frac{\Pr(x\not\in[z, z+\delta) \cap x\not\in[0, z))}{\Pr(x\not\in[0,z))}= \frac{\Pr(x\in[z+\delta, \infty))}{\Pr(x\not\in[0,z))}$$ But then: $$1-\Pr(x\not\in[z, z+\delta)|x\not\in[0,z))=\frac{\Pr(x\in[0,z+\delta))}{\Pr(x\not\in[0,z))}\neq \frac{\Pr(x\in[z, z+\delta))}{\Pr(x\not\in[0,z))}$$
Does anyone know what went wrong here? I can't seem to identify the error. Thank you.
It might be helpful to shorten the names of the probabilities, to make things less mentally taxing: $$a=\Pr(x\in[0,z))$$ $$b=\Pr(x\in[z,z+\delta))$$ $$c=\Pr(x\in[z+\delta,\infty))$$ Then the initial conditional probability is $\frac b{b+c}$, and the complement is $\frac c{b+c}$, and you correctly calculate both. But then in taking $1-\frac b{b+c}$ you incorrectly take this to be $\frac{1-b}{b+c}$, not $\frac{b+c-b}{b+c}$.