Axioms of Probability
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The document discusses the axioms of probability and some basic properties. It defines three axioms for assigning probability values to events in a finite sample space: 1) a probability is between 0 and 1, 2) the probability of the entire sample space is 1, and 3) for mutually exclusive events, the total probability is the sum of the individual probabilities. It then modifies the third axiom for infinite sample spaces and lists five elementary properties that can be obtained from the axioms, such as the probability of the complement of an event being 1 minus the original probability. The document also gives an example problem calculating the probability of not getting a white ball from a bag.
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Axioms of Probability
- 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