Verified Solution • 2025–26 Session moderate

QJEEM20Mad0cd: JEE Main 2020 Mathematics - Class 12 JEEM20Mad0cd Solution: Let E C denote the complement of an event E. Let E 1 , E 2 and…

Let EC denote the complement of an event E. Let E1 , E2 and E3 be any pairwise independent events with P(E1) > 0

and P(E1 ∩\cap E2 ∩\cap E3) = 0.

Then P( E2C∩E3C/E1E_2^C \cap E_3^C/{E_1} ) is equal to :

Multiple Choice Options

  • A P(E3C)P\left( {E_3^C} \right) - P(E2)
  • B P(E2C)P\left( {E_2^C} \right) + P(E3)
  • C P(E3C)P\left( {E_3^C} \right) - P(E2C)P\left( {E_2^C} \right)
  • D P(E3) - P(E2C)P\left( {E_2^C} \right)

Step-by-Step Verified Solution

Given E1 , E2 , E3 are pairwise indepedent events

so P(E1 ∩\cap E2 ) = P(E1 ).P(E2 )

and P(E2 ∩\cap E3 ) = P(E2 ).P(E3 )

and P(E3 ∩\cap E1 ) = P(E3 ).P(E1 )

and P(E1 ∩\cap E2 ∩\cap E3) = 0

Now P(E2C∩E3CE1)P\left( {{{E_2^C \cap E_3^C} \over {{E_1}}}} \right)

=P[E1∩(E2C∩E3C)]P(E1)= {{P\left[ {{E_1} \cap \left( {E_2^C \cap E_3^C} \right)} \right]} \over {P\left( {{E_1}} \right)}}

=P(E1)−[P(E1∩E2)+P(E1∩E3)−P(E1∩E2∩E3)]P(E1)= {{P\left( {{E_1}} \right) - \left[ {P\left( {{E_1} \cap {E_2}} \right) + P\left( {{E_1} \cap {E_3}} \right) - P\left( {{E_1} \cap {E_2} \cap {E_3}} \right)} \right]} \over {P\left( {{E_1}} \right)}}

=P(E1)−P(E1).P(E2)−P(E1).P(E3)−0P(E1)= {{P\left( {{E_1}} \right) - P\left( {{E_1}} \right).P\left( {{E_2}} \right) - P\left( {{E_1}} \right).P\left( {{E_3}} \right) - 0} \over {P\left( {{E_1}} \right)}}

=1−P(E2)−P(E3)= 1 - P\left( {{E_2}} \right) - P\left( {{E_3}} \right)

=1−P(E3)−P(E2)= 1 - P\left( {{E_3}} \right) - P\left( {{E_2}} \right)

=P(E3C)−P(E2)= P\left( {E_3^C} \right) - P\left( {{E_2}} \right)

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