QJEEM18Ma0198: JEE Main 2018 Mathematics - Class 12 JEEM18Ma0198 Solution: Let A, B and C be three events, which are pair-wise independent …

Question

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Let A, B and C be three events, which are pair-wise independent and

\overrightarrow E

denotes the completement of an event E. If

P\left( {A \cap B \cap C} \right) = 0

and

P\left( C \right) &gt; 0,

then

P\left[ {\left( {\overline A \cap \overline B } \right)\left| C \right.} \right]

is equal to :

Accepted Answer

Here,

P\left( {\overline A \cap \overline B \left| C \right.} \right) = {{P\left( {\overline A \cap \overline B \cap C} \right)} \over {P\left( C \right)}}

=

{{P\left[ {\left( {\overline {A \cup B} } \right) \cap C} \right]} \over {P\left( C \right)}}

=

{{P\left[ {C - \left( {A \cup B} \right)} \right]} \over {P\left( 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)} \over {P\left( C \right)}}

=

{{P\left( C \right) - P\left( {A \cap C} \right) - P\left( {B \cap C} \right)} \over {P\left( C \right)}}

(

\because

\left. {P\left( {A \cap B \cap C} \right) = 0} \right)

=

{{P\left( C \right) - P\left( A \right).P(C) - P\left( B \right).P(C)} \over {P\left( C \right)}}

[

\because

A, B and C are independent events]

= 1 - P(A) - P(B)

=

P\left( {\overline A } \right)

- P(B) or

P\left( {\overline B } \right)

- P(A)

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