Introduction to Sample Space and Probability - Random Signals and Processes Primer with MATLAB - page 1

Chapter 1
Introduction to Sample Space and Probability

1.1 Sample Space and Events

Probability theory is the mathematical analysis of random experiments [KLI86,
p. 11]. An experiment is a procedure we perform that produces some result or
outcome [MIL04, p. 8].
An experiment is considered random if the result of the experiment cannot be
determined exactly. Although the particular outcome of the experiment is not
known in advance, let us suppose that all possible outcomes are known.
The mathematical description of the random experiment is given in terms of:

￿ Sample space
￿ Events
￿ Probability
The set of all possible outcomes is called sample space and it is given the symbol
S . For example, in the experiment of a coin-tossing we cannot predict exactly if
“head,” or “tail” will appear; but we know that all possible outcomes are the
“heads,” or “tails,” shortly abbreviated as H and T, respectively. Thus, the sample
space for this random experiment is:

S ¼f H ; T g: (1.1)

Each element in S is called a sample point , s i . Each outcome is represented by a
corresponding sample point. For example, the sample points in ( 1.1 ) are:

s 1 ¼ H ; s 2 ¼ T : (1.2)

When rolling a die, the outcomes correspond to the numbers of dots (1-6).
Consequently, the sample set in this experiment is:

S ¼f 1 ; 2 ; 3 ; 4 ; 5 ; 6 g: (1.3)