Let \( E^{c} \) denote the complement of an event \( E \) . Let \( E, F, G \) be pairwise independent events with \( P(G)>0 \) and \( P(E \cap F \cap G)=0 \) . The \( P\left(E^{c} \cap F^{c} \cap \mid G\right) \) equals
We have \[ E \cap F \cap G = \emptyset \] \[ P(E^c \cap F^c \mid G) = \frac{P(E^c \cap F^c \cap G)}{P(G)} \] \[ = \frac{P(G) - P(E \cap G) - P(G \cap F)}{P(G)} \] From the Venn diagram: \[ E^c \cap F^c \cap G = G - E \cap G - F \cap G \] \[ = \frac{P(G) - P(E) P(G) - P(G) P(F)}{P(G)} \] (Since \(E, F, G\) are pairwise independent) \[ = 1 - P(E) - P(F) = P(E^c) - P(F) \]
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