Question

Let A, B, C be pairwise independent events with P(C) > 0 and $P\left(A\cap B\cap C\right)=0.\mathrm{If}P\left(\left(\overline{A}\cap \overline{B}\right)/C\right)=1-x,$ then x = ____________.

Answer 1

For given sets, A, B and C with P(C) > 0, P(A∩B∩C) = 0 and A, B, C are pairwise independent.
By deffination $P\left(\frac{A^c\cap B^c}{C}\right)=P\frac{\left(A^c\cap B^c\cap C\right)}{P\left(C\right)}$
$P\left(A^c\cap B^c\cap C^c\right)=P\left(C\right)-P\left(A\cap C\right)-P\left(B\cap C\right)+P\left(A\cap B\cap C\right)\left(\mathrm{using}\mathrm{Venn}‐\mathrm{diagram}\right)$



$\begin{array}{rcl}\therefore P\left(\frac{A^c\cap B^c}{C}\right)& =& \frac{P\left(C\right)-P\left(A\cap C\right)-P\left(B\cap C\right)+P\left(A\cap B\cap C\right)}{P\left(C\right)}\\ & =& \frac{P\left(C\right)-P\left(A\cap C\right)-P\left(B\cap C\right)}{P\left(C\right)}\left[\because P\left(A\cap B\cap C\right)=0\right]\\ \mathrm{i}.\mathrm{e}& =& \frac{P\left(C\right)-P\left(A\right).P\left(C\right)-P\left(B\right).P\left(C\right)}{P\left(C\right)}\left[\because \left(A,C\right)\mathrm{and}\left(B,C\right)\mathrm{pair}\mathrm{are}\mathrm{also}\mathrm{independent}.\right]\\ & =& \frac{P\left(C\right)\left[1-P\left(A\right)-P\left(B\right)\right]}{P\left(C\right)}\left[\mathrm{Since}P\left(C\right)>0\right]\\ \mathrm{i}.\mathrm{e}P\left(\frac{A^c\cap B^c}{C}\right)& =& 1-P\left(A\right)-P\left(B\right)\\ & =& 1-x\left(\mathrm{given}\right)\end{array}$
∴ x = P(A) + P(B)