CHAPTER 1
Probability
The idea around probability, chance and possibility is quite old and is commonly used in
everyday speech. For example, what is the possibility that it will rain today, or what is the
chance that your rugby team will win the match on Saturday? In everyday life these chances
and possibilities are based on peoples intuition based guesses.
Probability is a concept founded on solid mathematical theory, and can be a powerful aide in
important decision making. In modern society, daily decisions are made in almost every field
where there is some element of risk, such as in those involving large quantities of money, or
those involving human lives. In these situations accurate probability calculations are of
paramount importance (Swanepoel and Allison, 2011).
Probability theory plays an important role in Statistical Inference. In some situations only
sample data is available, in such cases probability theory in used to evaluate the conclusions
made about the population. Probability theory involves experiments in which the outcomes
occur randomly.
1.2 Sample spaces
Experiment: An experiment is an act where the outcome cannot be foreseen (predicted),
meaning that the result are not always the same and the results are not known in advance
before the experiment is conducted.
Sample space: The sample space of an experiment is a collection of all possible outcomes and
is denoted by ( ).
Event: An event (denotes by capital letters such as A, B, C, etc.) is a collection of some of the
outcomes.
Example 1
i) A fair coin is tossed:
ii) A dice is rolled:
Let A be the event that a number larger than 3 is rolled: A = {4,5,6}, .
iii) Two fair coins are tossed:
Let A be the event that at least one head is tossed: A = {HH,HT,TH}
Let B be the event that at least one tail is tossed: B = {TT,HT,TH}
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, See Example A and B (Rice, p. 2 - 3).
We make use of Venn-diagrams to illustrate certain concepts of probability theory.
A
Union: The union of two event A and B is the event that either A or B or both will take place,
denoted by .
Intersection: The intersection of two events A and B is the event that both A and B will take
place, denoted by .
Complement: The complement of an event A is the event that A will not take place denoted
by Ac or . Therefore all the outcomes in that are not in A.
Empty set: An empty set occur when the event is impossible, denoted by .
Disjoint: If two events A and B are disjoint (mutually exclusive) it means that if event A
occur event B cannot occur and vice versa, therefore .
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