Axioms of Probability

Axioms, elementary properties of probability , examples

1. Probability

2. Axioms of Probability Let S be a finite sample space and A be an event in S. Then in the axiomatic definition, the probability P ( A ) of the event A is a real number assigned to A which satisfies the following three axioms: 1. Axiom 1 : P ( A )  1 2. Axiom 2 : P(S) = 1 3. Axiom 3 : P ( A  B) = P ( A ) + P(B) if A  B = 

3. If the sample space S is not finite, then axiom 3 must be modified as follows: Axiom 3*: If , , , . . . is an infinite sequence of mutually exclusive events in Sample space S (  =  for i ≠ j), then  

4. Elementary Properties of Probability: By using the above axioms, the following useful properties of probability can be obtained: P(A’)= 1 – P(A) P( ) = 0 P(A)  P(B) if A  B  P(A)  1 P ( A  B) = P (A) + P(B) – P( A  B)

5. Example: The probability of getting a white ball from the bag of balls is ¼. What is the probability of not getting a white ball? Solution: Let A be the event of getting a white ball. A’ = event of not getting a white ball. P(A’)= 1 – P(A) = 1 – ¼ = ¾