M2L3. Axioms of Probability

Transcription

1 M2L3 Axioms of Probability 1. Introduction This lecture is a continuation of discussion on random events that started with definition of various terms related to Set Theory and event operations in previous lecture. Details of axioms of probability, their properties, examples and a brief on conditional probability are discussed in this lecture. Though the axiomatic definition probability was presented in lecture 1, these are mentioned again before proceeding to the elementary properties. 2. Axioms Probability of any event,, is assigned in such a way that it satisfies certain conditions. These conditions for assigning probability are known as Axioms of probability. There are three such axioms. All conclusions drawn on probability theory are either directly or indirectly related to these three axioms. Axiom 1. For any event belongs to the sample space,, the value of probability of the event lies between zero and one. Mathematically expressed as:. Thus, Axiom 1 states that probabilities of events for a particular sample space are real numbers on the interval [0, 1]. Axiom 2. Probability of all the events in a sample space or the sample space in total is equal to one. Mathematically denoted as,.

2 Axiom 3. For any two mutually exclusive events and, the probability of union of them is equal to simple sum of the probabilities of individual events. It is mathematically denoted as:. 3. Elementary Properties Property 1. If are mutually exclusive events, then probability of union of all these events is equal to summation of probability of individual events. This is mathematically denoted as, (1) The Venn diagram is shown below (Fig. 1). Fig. 1. Explanation of Elementary Property 1 through Venn diagram This is basically the extension of Axiom 3, considering any number of mutually exclusive events. This is known as property of infinite additivity. Property 2. If any event belong to, then probability of, will be less than or equal to probability of,. And the probability of difference between these two events, will be equal to the difference between probability of and, i.e., and.

3 This is mathematically denoted as: if, then and P A A P A P A2 (Fig. 2).. The visualization is given below in Venn diagram Fig. 2. Explanation of Elementary Property 2 through Venn diagram Property 3. If any event is complementary to another event, then probability of can be determined by probability of from Axiom 1. This is mathematically denoted as, if then. Visualization is given below (Fig. 3). Fig. 3. Explanation of Elementary Property 3 through Venn diagram Property 4. If an event, A is the union of events, where are mutually exclusive, then probability of is the summation of probability of each of these events.

4 This mathematically denoted as, if where, A, A, 1 2, An are mutually exclusive events, then (2) The visualization is given below (Fig. 4). Fig. 4. Explanation of Elementary Property 4 through Venn diagram Property 5. For any two events, and belong to sample space, probability of, can be determined by summation of probability of intersection and the same of and complement of. It is mathematically denoted as: (3) The visualization is given in Fig. 5. Fig. 5. Explanation of Elementary Property 5 through Venn diagram

5 Property 6. If and are any two events in sample space,, then probability of union of and can be determined by deducting the probability of intersection of and from the summation of individual probabilities. It is mathematically denoted as: The visualization is given in Fig. 6. (4) Fig. 6. Explanation of Elementary Property 6 through Venn diagram This property can be explained well by proving it using axioms and other properties. From Fig. 9, considering the different parts of the shaded areas, Extending this property, if,, are any three events, This can be visualized graphically in Fig. 7. Fig. 7. Proof of Extension of Property 6

6 Property 7. If an event results in occurrence of one of the mutually exclusive events, in the sample space, then probability of is equal to the sum of probabilities of intersection between and any event among. It is mathematically denoted as, The visualization is presented in Fig. 8. (5) Fig. 8. Explanation of Elementary Property 7 through Venn diagram Problem: If a steel-section manufacturer produces a particular section and initial quality check reveals that the probability of producing a defective unit is. Further investigation reveals that the probability of producing a defective unit in terms of measurement is, whereas, in terms of material quality is. What is the probability of producing a unit that is defective in measurement as well as quality? Answer: Suppose, is the event of production of a defective unit in terms of measurement and is the event of production of a defective unit in terms of material quality. As per the statement of the problem, the events and are mutually exclusive. So,, and. Thus using the statement of Property 6, we can calculate:

7 . 4. Conditional Probability If and are two events in sample space and, then the probability of given than has already occurred is denoted as and mathematically expressed as: P B B P A B A (6) P The explanatory Venn diagram is given in Fig. 9. Fig. 9. Venn diagram for Conditional Probability Problem: On a national highway, a stretch of 10km is declared as accident prone. Over this stretch, probability of accident at any location is equally likely. In the middle of the stretch, there is a bridge of length m. Given that an accident has occurred within first 6km stretch, what is the probability that it has occurred on the bridge? Answer: Suppose, is the event of accident occurred within first km of the stretch and is the event of accident occurred on the bridge. So, and Thus using expression for conditional probability,

8 5. Concluding Remarks Before finishing this lecture, let us summarize the important learning here. The three Axioms define the basic properties of probability of events in a sample space. The elementary properties formulated from the Probability Theorem, explain the probabilities of a particular event when other events exist in the same space are basically derived from the Axioms. The special probability theorems based on these elementary properties will be discussed in the next lecture.