The document discusses the classical definition of probability as well as axioms that define probability mathematically. It introduces the classical definition where probability is defined as the number of favorable outcomes divided by the total number of possible outcomes. It then discusses limitations of the classical definition and introduces the frequency interpretation of probability. Finally, it outlines three axioms that define a function as a valid probability function: 1) probabilities are between 0 and 1, 2) the probability of the sample space is 1, and 3) probabilities of mutually exclusive events sum to the total probability.

1. 1.2 Probability and its Axioms

2. Probability The classical probability concept If there are n equally likely possibilities, of which one must occur and s are regarded as favorable, or as a “success”, then the probability of a “success” is given by s / n. Ex: If a card is drawn from a well shuffled deck of 52 playing cards, then find probability of drawing (a) a red king, (b) a 3, 4, 5 or 6, (c) a black card (d) a red ace or a black queen. Ans: (a) 1/26, (b) 4/13, (c) 1/2, (d) 1/13.

3. Probability A major shortcomingof the classical probability concept is its limited applicability. There are many situations in which the various possibilities cannot all be regarded as equally likely. For example, if we are concernedwith the question of whether it will rain the next day, whether a missile launching will be a success, or whether a newly designed engine will function for at least 1000 hours.

4. Probability The frequency interpretation of probability: The probability of an event (or outcome) is the proportion of times the event would occur in a long run of repeated experiments. If the probability is 0.78 that a plane from Mumbai to Goa will arrive on time, it means that such flights arrive on time 78% of the time.

5. Probability if weather service predicts that there is a 40% chance for rain this means that under the same weather conditions it will rain 40% of the time. In the frequency interpretation of probability, we estimate the probability of an event by observing what fraction of the time similar event have occurred in the past.

6. The Axioms of probability We define probabilities mathematically as the values of additive set functions. f : A  B, A : domain of f If the elements of the domain of the function are sets, then the function is called Set function. Ex: Consider a function n that assigns to each subset A of a finite sample space S the number of elements in A, i.e.

7. The Axioms of probability A set function is called additive if the number which it assigns to the union of two subsets which have no element in common is sum of the numbers assigned to the individual subsets. In above example n is additive set function; that is

8. The Axioms of probability Let S be a sample space, let C be the class of all events and let P be a real-valued function defined on C. Then P is called a probability function and P(A) is called the probability of event A when the following axioms hold: Axiom 1 0  P(A) 1 for each event A in S. Axiom 2 P(S) = 1. Axiom 3 If A and B are mutually exclusive events in S, then P(AB) = P(A) + P(B).

9. The Axioms of probability Ex: If an experiment has the three possible and mutually exclusive outcomes A, B and C, check in each case whether the assignment of probabilities is permissible: P(A) = 1/3, P(B) = 1/3 and P(C) = 1/3; P(A) = 0.64, P(B) = 0.38 and P(C) = –0.02; P(A) = 0.35, P(B) = 0.52 and P(C) = 0.26; P(A) = 0.57, P(B) = 0.24 and P(C) = 0.19. Ans:a) Y, b) N, c) N, d) Y