an event. (iii) Mutually Exclusive events: Two events A and B are said to be mutually exclusive events if A i.e., if A and B are disjoint sets. Example: Consider S = { , , , , } Let A = the set of odd numbers = { , , } and B = the set of even numbers = { , } Then A ∩ B = z (iv) Therefore the events A and B are mutually exclusive. (xi) Observation: Statement meaning in terms of Set theory approach.
- - (i) A B & at least one of the events A or B occurs (ii) A B & both events A and B occurs (iii) A B & Neither A nor B occur (iv) A B & Event A occurs and B does not occur (xii) Definition of Probability (Axiomatic approach) Let E be an experiment. Let S be a sample space associated with E . With every event in S we associate a real number denoted by P ( A ) called the probability of the event A satisfying the following axioms. Axiom : P(A) ≥ Axiom : P(S) = Axiom : If A A A n … be a sequence of n mutually exclusive events in S then P A P A P A P A … ) = ( ) + ( ) +…+ ( (xiii) Basic Theorems on probability Theorem : P ( ∅ ) = i.e., probability of an impossible event is zero.
Theorem : Let S be the sample space and A be an event in S, then P( A ) = – P ( A ). Theorem : Addition Theorem If A and B are any two events, then P A P A P B P A ∪ ∩ ) = ( ) + ( ) − (xiv) Observation: (i) If the two events A and B are mutually exclusive, then A ∩ B = ∅ ∴ P ( A ∩ B ) = ⇒ P ( A B ) = P ( A )