Wednesday, October 29, 2008

Probability Of An Event

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

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