## 3.3 The Probability Measure

To create a mathematical model which will enable us to quantify the likelihood of the various outcomes of the experiment, we attach to each event a number between 0 and 1 which will be called the probability of the event. It can be thought of as the relative frequency of occurrence of the event in a large number of repetitions of the experiment. The probability of event A will be denoted by P( A). The probability measure satisfies the following axioms:

1. 0 ? P( A) ? 1 for all P( A).

2. P( S) = 1 and P( ) = 0.

3. If A ? B = (i.e. A and B are mutually exclusive), then P( A ? B) = P( A) + P( B).

It follows immediately from the axioms that if { A 1, A 2, , A n} is a partition then

Examples

1. Consider a room of fire origin with one door and one window. There are four possible situations, namely: door and window closed, door and window open, door open and window closed, and door closed and window open. By analysing observations on actual fires, we may come to the conclusion that the relative frequencies of these four events are 0.1, 0.2, 0.4 and 0.3.

2. Returning to the maximum temperature T in...

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##### Topics of Interest

3.4 Conditional Probability Definition 3.4.1 For any two events A and B such that P( B) ? 0 , the conditional probability of event A given that event B has occurred is denoted by P( A B) and is...

B.1 Probability Theory Probability theory is what is known as an axiomatic system. The basic idea is to define a collection of objects or sample space with each element being a sample point. A set of...

1.4 Axioms and Properties of Probability In this section, we present Kolmogorov s axioms and derive some of their consequences. The probability models of the preceding section suggest the following...

3.6 The Concept of Independence Intuitively, if event B has no effect on event A we can say that A is independent of B. More precisely, if knowledge that event B has occurred does not affect the...

Overview Probabilistic thinking is based on very old ideas which go back to De Mere [1], La Place [2], and Bayes [3]. What is probability and how does it relate to frequency, statistics and, finally,...

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