Probability Of An Event

RESULT 1

Consider the set S of all possible outcomes of an experiment or a trial. A set of this kind sometimes called a Probability space or Sample space. Any event A can be represented by the subset of S, which contains all the outcomes in which the event occurs

Let S contains a finite number of equally likely outcomes, say m, so that n(S) = n. Let the event A has m sample points so that n(A) = m.

We have

       P(A) = n(A)/ n(S) = m/n

                  = Number of Favourable Outcomes

                      Number of Possible Outcomes

                                        

              Since A is a subset of S,

             Therefore 0 ≤ m ≤ n  i.e., 0 ≤ m/n ≤ 1

                                            Hence 0 ≤ P(A) ≤ 1

Note: The probability of an event A is a number between 0 and 1 inclusive. If  P(A) = 0, then the event cannot possibly occur.

If  P(A) = 1, then the event is certain to occur.

RESULT 2       

       _

Let  A denote the event  'A does not occur' 

                              _           _

                 Now P(A)  = n(A) /n(S) = n-m = 1-m     

                                                                  n           n

                                                                 = 1 -  P(A)

                                                 _       

                             Therefore P(A) = 1 -  P(A)

                                                              _

                            This implies P(A) + P(A) = 1