Page 199 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes

Task Define disjoint events with example.

Self Assessment

Fill in the blanks:

7. Frequently we are not concerned in a single result but in whether or not one of a group of
results appears. Such subsets of the sample space are known as ............................. .
8. We say that event A appears if the result of the experiment is one of the .............................
in A.
9. Two events A and B which have no results in general, that is, A  B = , are known as
............................. .
10. If A  B (A is a subset of B) then event A is said to ............................. event B.

c


11. If {Ai} is a compilation of events (sets) then  A i  ............................. .
 i 
14.4 Probability

The third element in the model for a random experiment is the requirement of the probability
of the events. It informs us how likely it is that a specific event will take place.

Definition

A probability P is a rule which allocates a positive number to every event, and which assures the
following axioms:

Axiom 1: P(A)  0.
Axiom 2: P() = 1.
Axiom 3: For any series A1,A2, . . . of disjoint events we have

 
i  
P  A  P   .A i ...(3)
 i  i
Axiom 2 just specifies that the probability of the “certain” event  is 1. Property (1.3) is the vital
property of a probability, and is sometimes known as the sum rule. It just specifies that if an
event can occur in a number of different manners that cannot occur simultaneously then the
probability of this event is just the sum of the probabilities of the composing events.
Note that a probability rule P has precisely the similar properties as the general “area measure”.

Example: The whole area of the union of the triangles in Figure 14.9 is equivalent to the
sum of the areas of the individual triangles. This is how you should understand property (3). But
rather than gauging areas, P computes probabilities.
As a direct effect of the axioms we have the following properties for P.

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