Let E^c denotes the complement of an event E. If E, F, G… — JEE Main 2007 Mathematics PYQ

AIEEE 2007

Let

E^c

denotes the complement of an event

E

. If

E

F

G

are pairwise independent events with

P(G) > 0

and

P(E \cap F \cap G) = 0

. Then,

P(E^c \cap F^c | G)

equals

A.
$P(E^c) + P(F^c)$
B.
$P(E^c) - P(F^c)$
C.
$P(E^c) - P(F)$
D.
$P(E) - P(F^c)$

Show answer and solution

Correct option: C

Solution

P(Eᶜ ∩ Fᶜ | G) = P(Eᶜ ∩ Fᶜ ∩ G) / P(G).

Now P(Eᶜ ∩ Fᶜ ∩ G) = P(G) - P(E ∩ G) - P(F ∩ G) + P(E ∩ F ∩ G).

Since E,F,G are pairwise independent: P(E ∩ G) = P(E)P(G) and P(F ∩ G) = P(F)P(G), and P(E ∩ F ∩ G) = 0.

So P(Eᶜ ∩ Fᶜ | G) = [P(G) - P(E)P(G) - P(F)P(G)] / P(G) = 1 - P(E) - P(F) = P(Eᶜ) - P(F), which is option 3.

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