How can I prove randomness

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»Event operations on the Venn diagram
»\ ((...), \ {... \} \) and other spellings
»Venn diagram for \ (A, B \) and \ (C \)

Basic concepts

In the following we always examine a random experiment. The random experiment is often used synonymously. A chance experiment is the observation of an experiment, the outcome of which is unknown to us and acts randomly. Classic examples are the dice toss, the coin toss, the game of roulette, but also the color of the traffic light when you turn into a street, or the gender of a child are to a certain extent random.

All possible test results are summarized under the basic set \ (\ Omega \) (Omega, the last letter of the Greek alphabet, from \ (A \) to \ (O \), from beginning to end, from Alpha to Omega), at Dice roll would be \ (\ Omega_W = \ {1,2,3,4,5,6 \} \) and at the traffic lights \ (\ Omega_A = \ {Ge, Ro, Gr \} \) for money, red and Green. (Rarely) special cases such as "dump" or "broken traffic light" are usually excluded. We call the individual events of \ (\ Omega \) elementary events \ (\ omega_1, \ dots \ omega_n \).

A possible event \ (A \) (or test outcome) is then a subset of \ (\ Omega \), one writes \ (A \ subset \ Omega \). So the event \ (A: = \) "a 1 is rolled" is very clear \ (A = \ {1 \} \ subset \ {1,2,3,4,5,6 \} = \ Omega_W \) . But we can also look at complicated events, for example \ (B: = \) "a prime number is rolled". This then results
\ begin {align *}
B = \ {2,3,5 \} \ subset \ {1,2,3,4,5,6 \} = \ Omega_W.
\ end {align *}
The event \ (B \) therefore consists of the three elementary events 2, 3 and 5.

Every event has a counter-event \ (A ^ c \), \ (c \) of "complement", often \ (A '\) or \ (\ bar {A} \) is also written. Colloquially it consists of "all other elements of \ (\ Omega \)" and we write \ (\ Omega \ setminus A \) (Omega without \ (A \)). In the previous example we get for
\ begin {align *}
A ^ c = \ {2,3,4,5,6 \} = \ {1,2,3,4,5,6 \} \ setminus \ {1 \}
\ end {align *}

Event operations on the Venn diagram

In this section we would like to illustrate the various operations between sets on a Venn diagram. The Venn diagram consists of two subsets \ (A, B \) of \ (\ Omega \), i.e. \ (A, B \ subset \ Omega \). Usually it looks like this:

For example, the middle area intuitively describes the elementary events that belong to \ (A \) and \ (B \). The set \ (A ^ c \) can be described as follows:

And now let's look at other operations. The previously mentioned intersection \ (A \ cap B \) of two sets describes the elements that are present in \ (A \) and \ (B \).

The union \ (A \ cup B \) denotes the elements which are contained in \ (A \) or \ (B \). "Or" does not mean "either or" here, the elements of \ (A \ cup B \) must therefore be contained in at least one of the two sets \ (A, B \).

If we want to describe \ (A \) without the elements of \ (B \), we write \ (A \ setminus B \).

If the sets \ (A \) and \ (B \) have nothing in common, so \ (A \ cap B = \ emptyset \), the whole thing looks like this:

\ ((...), \ {... \} \) and other spellings

Ideally, an event is clearly defined, for example \ (U \) is the event of throwing different symbols when tossing a coin twice. We then write \ (U: = \) "different symbols when tossing a coin twice". Often, however, you want to save time and make do with brackets. So one understands by \ ((W, K) \) the event of first throwing a coat of arms, then a head. The round brackets \ ((...) \) therefore indicate an order. On the other hand, \ (\ {... \} \) don't do this and \ (\ {W, K \} \) means once coat of arms, once head, regardless of the order.

Venn diagram for \ (A, B \) and \ (C \)

Venn diagrams can also be created for three subsets \ (A, B, C \).

For \ (A \ cap B \ cap C \) then applies

From then on it becomes difficult, however, the geometrical considerations for two and three sets can also be generalized for more sets.