Set Theoretical Notations of Probability - Practice Questions & MCQ

The Addition Rule of Probability 

(Probability of the event ‘A or B’ )

If A and B are any two events defined on a sample space, then the probability of occurrence of at least one of the event A and B is P(A ∪ B) and it equals P(A) + P(B) - P(A ∩ B) .

Special case

If A and B are disjoint sets, i.e., they are mutually exclusive events, then A ∩ B = φ. Therefore P(A ∩ B) = P (φ) = 0

Thus, for mutually exclusive events A and B, we have P(A ∪ B) = P(A) + P(B), 

If A and B are two events, then

             (A - B) ∩ (A ∩ B) = φ     and      A = (A - B) ∪ (A ∩ B) 

So,        P(A) =  P(A - B) + P(A ∩ B) - 0

                     =  P(A ∩ B’) + P(A ∩ B) 

or    P(A) - P(A ∩ B) =  P(A ∩ B’) = P(A - B)                   [∵ A - B = A ∩ B’ ]

Similarly, P(B) - P(A ∩ B) =  P(B ∩ A’) = P(B - A)  

If A, B and C are any three events in a sample space S, then

P(A ∪ B ∪ C) = P(A) + P(B) + P(C) -  P(A ∩ B) - P(A ∩ C) - P(B ∩ C) + P(A ∩ B ∩ C)

If A, B and C are any three mutually exclusive  events in a sample space S, then

P(A ∪ B ∪ C) = P(A) + P(B) + P(C)

Probability of event ‘not A’ or Complementary Event

If E is the any event and E’ be the complement of the event E. Since, E and E’ are disjoint and exhaustive sets.