Independent Events and Their Important Properties JEE Notes | EduRev

Mathematics (Maths) Class 12

JEE : Independent Events and Their Important Properties JEE Notes | EduRev

The document Independent Events and Their Important Properties JEE Notes | EduRev is a part of the JEE Course Mathematics (Maths) Class 12.
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Independent Events

Two events A & B are said to be independent if occurrence or non occurrence of one does not effect the probability of the occurrence or non occurence of other.

(a) If the occurrence of one event affects the probability of the occurrence of the other event then the events are said to be dependent or Contingent. For two independent events A and B 

Independent Events and Their Important Properties JEE Notes | EduRev

. P(B). Often this is taken as the definition of independent events.

(b) Three events A, B & C are independent if & only if all the following conditions hold ;

Independent Events and Their Important Properties JEE Notes | EduRev

i.e., they must be pairwise as well as mutually independent.

Similarly for n events A1, A2, A3, ........... An to be independent, the number of these conditions is equal to nC2 + nC3 + ............. + nCn = 2n - n - 1.

Note : Independent events are not in general mutually exclusive & vice versa.

Mutually exclusiveness can be used when the events are taken from the same experiment & independence can be used when the events are taken from different experiments.

Ex.1 The probability that an anti aircraft gun can hit an enemy plane at the first, second and third shot are 0.6, 0.7 and 0.1 respectively. The probability that the gun hits the plane is

Sol. Let the events of hitting the enemy plane at the first, second and third shot are respectively A, B and C. Then as given P(A) = 0.6, P(B) = 0.7, P(C) = 0.1

Since events A, B, C are independent, so

Independent Events and Their Important Properties JEE Notes | EduRev

⇒ 

Independent Events and Their Important Properties JEE Notes | EduRev

    

Independent Events and Their Important Properties JEE Notes | EduRev

= 1 - (1 - 0.6) (1 - 0.7) (1 - 0.1)) = 1 - (0.4)(0.3)(0.9) = 1 - 0.108 = 0.892

 

Ex.2 If two events A and B are such that 

Independent Events and Their Important Properties JEE Notes | EduRev

, P(B) = 0.4 and 

Independent Events and Their Important Properties JEE Notes | EduRev

then 

Independent Events and Their Important Properties JEE Notes | EduRev

equals

Sol.

Independent Events and Their Important Properties JEE Notes | EduRev

 

 Probability of Three Events

Independent Events and Their Important Properties JEE Notes | EduRev

For any three events A, B and C we have

(a) P(A or B or C) = P(A) + P(B) + P(C) – P(A∩B) – P(B ∩ C) – P(C ∩ A) + P(A ∩ B ∩ C)

(b) P (at least two of A, B, C occur) = P(B ∩ C) + P (C ∩ A) + P(A ∩ B) – 2P (A ∩ B ∩ C)

(c) P (exactly two of A, B, C occur) = P(B ∩ C) + P (C ∩ A) + P(A ∩ B) – 3P (A ∩ B ∩ C)

(d) P (exactly one of A, B, C occurs) = P(A) + P(B) + P(C) – 2P (B ∩ C) – 2P (C ∩ A) – 2P (A ∩ B) + 3P (A ∩ B ∩ C)

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