I'm sure I've missed something simple here, but I'm working with a question of the form $$ P(A | \overline{B}) $$ Using the conditional law, this becomes $$ P(A | \overline{B}) = \frac{Pr(A \cap \overline{B})}{P(\overline{B})} $$ Here is the step that I'm unsure of. Here I thought I could apply the compliment law to the denominator and numerator, ie.
$$ P(A | \overline{B}) = \frac{1 - Pr(A \cap B)}{1 - P(B)} $$ Is this correct? When I apply it to a contingency table of data I'm working with, it yields the wrong answer.
Thanks!
For any two events $A$ and $B$, which are subsets of the population space $\Omega$,we have: $$\Pr[A] = \Pr[A \cap B] + \Pr[A \cap \overline{B}]$$ If we take $A = \Omega$ in the above equation we get $1 = \Pr[B] + \Pr[\overline{B}]$. Hence $$\Pr[A | \overline{B}] = \frac{\Pr[A \cap \overline{B}]}{\Pr[\overline{B}]} = \frac{\Pr[A] - \Pr[A \cap B]}{1 - \Pr[B]}$$