Let Ec denote the complement of an event E. Let E, F, G be pairwise independent events with P(G) 0 and P(E∩ F∩ G) = 0. Then P(Ec ∩ Fc | G) equals

Question

Let Ec denote the complement of an event E. Let E, F, G be pairwise independent events with P(G) > 0 and P(E∩ F∩ G) = 0. Then P(Ec ∩ Fc | G) equals (A) P(Ec) + P(Fc) (B) P(Ec) - P(Fc) (C) P(Ec) - P(F) (D) P(E) - P(Fc)

Tardigrade Admin · Accepted Answer

We have ∵ E∩ F∩ G=φ P(Ec∩ Fc/G)=(P(Ec∩ Fc∩ G)/P(G)) =(P(G)-P(E∩ G)-P(G∩ F)/P(G)) [From venn diagram Ec ∩ Fc ∩ G = G - E ∩ G - F ∩ G] =(P(G)-P(E)P(G)-P(G)P(F)/P(G)) [∵ E, F, G are pairwise independent] = 1- P (E) - P (F) = P (Ec) - P (F)

Q. Let $E^{c}$ denote the complement of an event $E$ . Let $E, F, G$ be pairwise independent events with $P (G) > 0$ and $P (E \cap F \cap G) = 0$ . Then $P (E^{c} \cap F^{c} ∣ G)$ equals

$P (E^{c}) + P (F^{c})$

$P (E^{c}) - P (F^{c})$

$P (E^{c}) - P (F)$

$P (E) - P (F^{c})$

Q. Let Ec denote the complement of an event E. Let E,F,G be pairwise independent events with P(G)>0 and P(E∩F∩G)=0. Then P(Ec∩Fc∣G) equals

P(Ec)+P(Fc)

P(Ec)−P(Fc)

P(Ec)−P(F)

P(E)−P(Fc)

We have ∵E∩F∩G=ϕ P(Ec∩Fc/G)=P(G)P(Ec∩Fc∩G)​ =P(G)P(G)−P(E∩G)−P(G∩F)​ [From venn diagram Ec∩Fc∩G=G−E∩G−F∩G] =P(G)P(G)−P(E)P(G)−P(G)P(F)​ [∵ E, F, G are pairwise independent] =1−P(E)−P(F)=P(Ec)−P(F)

Q. Let $E^{c}$ denote the complement of an event $E$ . Let $E, F, G$ be pairwise independent events with $P (G) > 0$ and $P (E \cap F \cap G) = 0$ . Then $P (E^{c} \cap F^{c} ∣ G)$ equals

$P (E^{c}) + P (F^{c})$

$P (E^{c}) - P (F^{c})$

$P (E^{c}) - P (F)$

$P (E) - P (F^{c})$