If overline{E} and overline{F} are the comple | vedclass

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(D) We know that for any event $A$ and a given event $B$ with $P(B) > 0$,the sum of conditional probabilities of an event and its complement given $B$

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$(a)$ and $(c)$ both

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(D) We know that for any event $A$ and a given event $B$ with $P(B) > 0$,the sum of conditional probabilities of an event and its complement given $B$ is $1$,i.e.,$P(A/B) + P(\overline{A}/B) = 1$.For option $(a)$: $P(E/F) + P(\overline{E}/F) = \frac{P(E \cap F)}{P(F)} + \frac{P(\overline{E} \cap F)}{P(F)} = \frac{P(E \cap F) + P(\overline{E} \cap F)}{P(F)}$. Since $(E \cap F)$ and $(\overline{E} \cap F)$ are disjoint and their union is $F$,the numerator is $P(F)$. Thus,$P(E/F) + P(\overline{E}/F) = \frac{P(F)}{P(F)} = 1$. So,$(a)$ is true.For option $(c)$: Similarly,$P(E/\overline{F}) + P(\overline{E}/\overline{F}) = \frac{P(E \cap \overline{F}) + P(\overline{E} \cap \overline{F})}{P(\overline{F})} = \frac{P(\overline{F})}{P(\overline{F})} = 1$. So,$(c)$ is true.Therefore,both $(a)$ and $(c)$ are correct.

If $\overline{E}$ and $\overline{F}$ are the complementary events of events $E$ and $F$ respectively and if $0 < P(F) < 1,$ then

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