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Basic concepts of probability theory

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Here you will get to know the basic concepts of probability calculation such as random experiment, result and event. You will also learn here how you can represent possible results of random experiments with the help of tree diagrams.

Recognizing random experiments

A random experiment is a process that can have more than one possible outcome (a so-called result). However, it cannot be predicted what result the random experiment will have.
coin toss Throwing a coin is a random experiment as both heads and tails can appear. However, it cannot be predicted whether heads or tails will appear.
roll the dice Throwing a die labeled with the numbers 1, 2, 3, 4, 5 and 6 is a random experiment, as any of the numbers mentioned can appear as a result. However, it cannot be predicted which of the six digits will appear.
Recognizing random experiments Which of the following processes are random experiments or which of the events described are random?

Result - event - result set

The possible outcomes of a random experiment are called results. If you summarize all possible results of a random experiment in a set, you get the result set. It is usually denoted by the symbol Ω (pronounced Omega). Any combination of one or more results of a random experiment in a set is called an event. There is also what is known as the "impossible event" that has no result. This is covered in the following section.
Results, events and result set for the cube Throwing a dice is a random experiment. Each of the possible numbers 1, 2, 3, 4, 5 and 6 is a result. The result set is the summary of all possible outcomes in a set. When rolling the dice, Ω = {1; 2; 3; 4; 5; 6} the result set. A possible event is, for example, “dice shows an even number” and is represented by the set {2; 4; 6} shown.
The results of a random experiment What is a possible result when throwing a dice once with the pictured net of dice?
The result set when tossing a coin When tossing a coin, the results heads (K) or tails (Z) can appear. Thus the result set is Ω = {K; Z}.
Which of the specified quantities is a result set for drawing a ball from the urn shown?
Events when rolling the dice A dice with the numbers 1, 2, 3, 4, 5 and 6 is thrown once. The event E - “A number smaller than 3 is rolled” - consists of the results 1 and 2. Usually such an event is written as a set: E = {1; 2}.

Safe and impossible events, as well as counter-events

For every event E of a random experiment there is a so-called counter event. It consists of all those possible results of the random experiment that do not belong to E. The result set Ω is the summary of all possible results of a random experiment. So it is also an event. Since this event always occurs, this event is also called a safe event. The counter-event to the certain event is characterized by the absence of possible results of a random experiment. So it is the empty set and is denoted by ∅ or {}. One calls ∅ or {} the impossible event.
A die with the numbers 1, 2, 3, 4, 5 and 6 is rolled once. Furthermore, let E be the event “A number greater than or equal to 3 is rolled”. That is, it is E = {3; 4; 5; 6}. Then the counter-event belonging to E consists of those numbers that do not belong to E. Thus = {1; 2}.
A die with the numbers 1, 2, 3, 4, 5 and 6 is rolled once. Furthermore, let E be the event “A positive number is rolled”. Then E is the sure event, because every number that can be rolled is positive. That is, it is E = {1; 2; 3; 4; 5; 6} = Ω.
Two balls are drawn from the depicted urn without being replaced and E is the event “At least one of the balls is red or blue”. Decide whether the described event E is certain, impossible or random.
Two balls are drawn from the urn shown without replacing and E is the event “Both balls are yellow”. Decide whether the described event E is certain, impossible or random.

Multi-level random experiments and tree diagrams

If several random experiments are carried out one after the other, one speaks of a multi-stage random experiment. Two or three-stage random experiments can be represented very well with the help of tree diagrams (trees for short), provided that the individual partial experiments do not have too many results. The structure of a tree diagram that belongs to a multi-level random experiment is explained here using a few examples:
Another correct tree diagram is this: Again, in addition to the end nodes of the tree diagram, the result of the multi-level random experiment belonging to the respective path can be added. In our example, a complete tree diagram for turning the left wheel of fortune followed by turning the right wheel of fortune, including all results, looks like this:
Instead of coloring the nodes and results accordingly, you can, for example, choose the first letters of the colors to mark the nodes and results. In our case (R - Red, G - Yellow, B - Blue) an associated tree diagram would look like this: Also note the spelling of the results as a concatenation of the first letters of the colors. One could have given the results as ordered pairs of the first letters of the colors. (R; G) would then be the result: "Left wheel of fortune remains on red - right wheel of fortune remains on yellow." Again, for more extensive calculations, you often only enter the results associated with a specific event. If you are only interested in the results associated with event E “The color yellow appears exactly once”, the tree diagram would look like this: