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 𝐴1, 𝐴2 , 𝐴3, . . . is an infinite sequence of mutually
exclusive events in Sample space S (𝐴𝑖  𝐴𝑗 =  for i ≠ j), then

4. Elementary Properties of Probability:
By using the above axioms, the following useful properties of
probability can be obtained:
1. P(A’)= 1 – P(A)
2. P() = 0
3. P(A)  P(B) if A  B
4. 0 P(A)  1
5. 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