General speaking, probability of an event is the ability of that event occurrence. It is expressed as a number between 0 and 1. For example, when we flipping a homogeneous coin, ability that the upper face is the head (the face with picture or icon on it) is 50%, then the probability of this event is 0,50.

Probability has the close relation with relative frequency. Thanks to this relation, we can estimate probability in some simple cases. For example, there are 3 red marbles and 7 blue marbles in a jar, all are homogeneous. The probability that we draw a red marble from that jar is 0,30.

 

In order to obtain values of a variable, we have to conduct “experiments”, realize observations, measurements. In ordinary scientific experiment, we can know or guess the result. For example, if we mix HCl and NaOH, we known that we will obtain NaCl and H₂O. Or if we let an object falling freely in the air, we know that the speed of this object increases gradually with acceleration of 9,81 m.s⁻².

But in an experiment in statistics, we can not know or guess the result. Two experiments, realized in the exactly similar conditions, can obtain different results. This type of experiment is known as “statistical experiment”, or “probability experiment”, or “random experiment”, or briefly “trial”.

The result of such experiment is defined as "outcome". For example when we roll the die, its upper face is an outcome.

 

Sample space is the set of all possible outcomes of a statistical experiment.

Example 1 : Tossing one coin, if the upper face has picture or icon on it, the outcome is symbolized as H (head). On the contrary (the upper face has number on it) the outcome is symbolized as T (tail). So the sample space consists of two elements and can be symbolized as:

    S = { H, T }

Example 2 : Rolling a die, outcome is defined as the upper face of die. Sample space consists of 6 elements and:

    S = { 1, 2, 3, 4, 5, 6 }

Example 3 : Tossing two coins 1 and 2. Outcome is defined as the upper faces and symbolized as AB, which A is the upper face of coin 1 and B is the upper face of coin 2. So sample space of this statistical experiment consists of 4 elements:

    S = { HH, HT, TT, TH }

Example 4 : Rolling two dice A and B. Outcome is defined as the upper faces and symbolized as (ab), which a is the upper face of die A and b is the upper face of die B. So sample space consists of 36 elements:

     S = { (11), (12), (13), (14), (15), (16),
           (21), (22), (23), (24), (25), (26),
           (31), (32), (33), (34), (35), (36),
           (41), (42), (43), (44), (45), (46),
           (51), (52), (53), (54), (55), (56),
           (61), (62), (63), (64), (65), (66) }

 

From the examples above, we can recognize that sample space is a set of elements whose values’s type are the same. Each element is an outcome of a statistical experiment, and each element is considered as an elementary event (or simple event). Hence, elementary event, denoted as E_`i`, consists of only one outcome.

In general, an event can consist of many outcomes, and can be defined of a set of outcomes. It means that event, denoted as E, is a subset of sample space. To illustrate this relation, we use Venn diagram as in Fig. 1. An event said “happen” when it satisfies the determined conditions.

Fig. 1 Venn diagram of sample space S, event E and elementary event E_`i`

Example 5 : Given Example 3, A is the event that one upper face is H. So A is the combination of two simple events HT and TH: A = { HT, TH }

Example 6 : Given Example 2, each value of the upper face of the die can be considered as an elementary event. Getting an even upper face is an event consists of 3 elements: { 2, 4, 6 }.

Example 7 : Given Example 3, the upper faces of two coins are similar is an event consists of 2 elements { HH, TT }.

Example 8 : pH = 4,2 is an elementary event. pH < 4,5 is an event consists of infinite elements.

From these examples, we can recognize that from one sample space, we can formulate conditions (or rules) to create a lot of events.