Some Basic Definitions

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# OVERVIEW OF THE FIRST MAIN CHAPTER #


This first chapter brings a brief summary of  probability theory, which surely many of you already know from your school days and which is an important prerequisite for understanding the chapters that follow.

This chapter includes

  • some  definitions  such as random experiment, outcome, event, and probability,
  • the  set-theoretical basics relevant for probability theory,
  • the clarification of  statistical dependence  or  statistical independence,
  • the mathematical treatment of statistical dependence by  Markov chains.


Experiment and Outcome


The starting point of any statistical investigation is a  random experiment. By this, one understands

  • an experiment that can be repeated as often as desired under the same conditions with an uncertain  outcome  $E$,
  • in which, however, the quantity  $ \{E_μ \}$  of the possible outcomes is specifiable.


$\text{Definition:}$  The number of possible outcomes is called the  range of outcomes  $M$.  Then holds:

$$E_\mu \in G = \{E_\mu\}= \{E_1, \hspace{0.1cm}\text{...} \hspace{0.1cm}, E_M \} .$$

The variable  $μ$  can take all integer values between  $1$  and  $M$ .  $G$  is called the event space or the  basic set.


$\text{Example 1:}$  In the experiment "coin toss" there are only two possible outcomes, namely "heads" and "tails"   ⇒   $M = 2$.  In contrast, in the random experiment "throwing a roulette ball" a total of  $M = 37$  different outcomes are possible, and it holds for the basic set in this case:

$$G = \{E_\mu\} = \{0, 1, 2, \text{...} \hspace{0.1cm} , 36\}.$$

Classical definition of probability


We first assume that each trial results in exactly one outcome from  $G$  and that each of these  $M$  outcomes is possible in the same way (without preference or disadvantage).

$\text{Definition:}$  With this assumption, the  probability  of each outcome is  $E_μ$  equally:

$$\Pr (E_\mu) = 1/{M}.$$


This is the  classical definition of probability.  ${\rm Pr}(\text{...} )$  stands for  probability  and is to be understood as a mathematical function.

$\text{Example 2:}$  In the random experiment "coin toss", the probabilities of the two possible outcomes are:

$$\rm Pr("Heads")=Pr("Tails")=1/2.$$

This assumes that each attempt ends either with "heads" or with "tails" and that the coin cannot come to rest on its edge during an attempt.

Also in the experiment "throwing a roulette ball" the probabilities  ${\rm Pr}( E_μ) = 1/37$  are equal for all numbers from  $0$  to  $36$  only if the roulette table has not been manipulated.


Note:   Probability theory – and the statistics based on it – can only provide well-founded statements if all implicitly agreed conditions are actually fulfilled. 

  • Checking these conditions is not the task of statistics, but of those who use them. 
  • Since this basic rule is often violated, statistics has a much worse reputation in society than it actually deserves.

Event and Event Space


$\text{Definitions:}$  By an  event  we mean a set or summary of outcomes.

  • We refer to the set of all events as thenbsp; event set  $\{A_i \}$.
  • Since the number  $I$  of possible events  $\{A_i \}$  is generally not the same as the number  $M$  of possible outcomes - that is, the elements of  $G = \{ E_μ \}$  – different indices are chosen here.

If an event  $A_i$  is composed of  $K$  (elementary) outcomes, the  event probability  is defined as follows:

$${\rm Pr} (A_i) = \frac{K}{M} = \frac{\rm Number\hspace{0.1cm}of\hspace{0.1cm}favorable\hspace{0.1cm}outcomes}{\rm Number\hspace{0.1cm}of\hspace{0.1cm}possible\hspace{0.1cm}outcomes}.$$


This equation is called the  Laplace probability definition.

  • Here, "favorable outcomes"are those outcomes that belong to the composite event  $A_i$ .
  • From this definition it is already clear that a probability must always lie between  $0$  and  $1$   (including these two limits).


$\text{Example 3:}$  We consider again the experiment "throwing a die". The possible outcomes are thus  $E_μ ∈ G = \{1, 2, 3, 4, 5, 6\}$.

Let us now define two events  $(I = 2)$, viz.

  • $A_1 = \big[$the number of dice is even$\big] = \{2, 4, 6\}$,  and
  • $A_2 = \big[$the number of eyes is odd$\big] = \{1, 3, 5\}$,


then the event set  $\{A_1, A_2\}$  is equal to the basic set  $G$.  For this example, the events  $A_1$  and  $A_2$  represent a so-called  complete system .

On the other hand, the further event set  $\{A_3, A_4\}$  is not equal to the basic set  $G$, if we define the single events as follows:

  • $A_3 = \big[$the number is smaller 3$\big] = \{1, 2\}$,
  • $A_4 =\big[$the number is bigger than 3$\big] = \{4, 5, 6\}$.


Here, the event set  $\{A_3, A_4\}$  does not include the element „3”.  The probabilities of the events defined here are  ${\rm Pr}( A_3) = 1/3$  and  ${\rm Pr}( A_1) ={\rm Pr}(A_2) = {\rm Pr}(A_4) = 1/2$.


The topic of this chapter is illustrated with examples in the learning video  Classical Definition of Probability .


Exercises for the chapter


Aufgabe 1.1: Würfelspiel Mäxchen

Aufgabe 1.1Z: Summe zweier Ternärsignale