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I hope you're having a wonderful day. Welcome to this new episode. If you want to talk about mathematics, assuming it’s a serious conversation, you need basic vocabulary. Therefore, today we’ll clarify a few terms that are fundamental for dealing with probability.

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An important objective is to define what is meant by probability. Naturally, you cannot be contented with an intuitive approach in mathematics. We won’t yet achieve this objective today, but we will prepare it.

Once again, we’ll select random experiments as the starting point. As you already know, the outcome of such an experiment cannot be predicted, but clearly one of certain, predetermined results will occur. These results are summarized in the set of outcomes symbolized with 𝜴.

Slide 2

And so we have already used two terms, specifically outcome and set of outcomes. How do you use these terms? And what actually is an outcome? Is a “6” shown after rolling a die an outcome? In colloquial terms, you would express it this way.

Let’s look at this step by step based on examples. In the process, the basic idea is that a random experiment has desired, thus defined and determined results. Casually, the outcomes are desired, defined, and determined by you as the person conducting the experiment.

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Here are two simple examples: first, flipping a coin and second, a soccer match. When you flip two coins, in principle there are three possibilities, which are shown here: 2 times heads, 2 times tails, or 1 time heads and 1 time tails. We could create a set of outcomes 𝜴 from these possibilities.

In the same way, there are three possible outcomes for a soccer match because either of the two teams can win or the match can end in a tie.

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If you roll a die, you may accordingly be interested in each of the six possible outcomes and determine the set of outcomes as 𝜴 = {1, 2, 3, 4 ,5, 6}.

However, imagine that you are playing “Trouble,” similar to the German game “Mensch ärgere dich nicht”. The only important thing at the beginning is whether you roll a 6. You can easily adapt the set of outcomes to this situation. It is 𝜴 = {6, not 6}, because all other numbers but 6 are equally unappealing.

Perhaps you are interested in even versus odd numbers; then take that as your set of outcomes.

Once again: The basic idea is that a random experiment has results determined by you as the person conducting the experiment.

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Here is another example. Let’s assume you are rolling two dice. Then you might be interested in what the two dice show without caring about the sequence. You thus end up with the set of outcomes 𝜴₁, consisting of all pairs of numbers from 1 to 6. Or you add the number of spots and determine whether the sum is even or odd. Then you determine the set of outcomes as 𝜴₂ . You can already tell that this game can go on for as long as you want.

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Here are a couple more of those brief examples. The first and second examples make it clear once again that it really depends on what is defined as the set of outcomes. The second and third examples show very explicitly that experiments can be assigned a set of events for which there are no theoretical assumptions for the frequency.

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And this is the definition: The set Ω = {ω₁, ω₂, ... , ω_n} of all possible results of a random experiment is called the sample space or set of outcomes.

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It should be noted that the set of outcomes must cover all possible outcomes of the random experiment. Otherwise, the definition would be simply pointless in practice.

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Precise language is important in mathematics, as I already said. Let us define more precisely the actions we just took. We consider solely discrete random experiments. These are experiments in which the set of outcomes 𝜴 is discrete. A discrete set has a finite number or a countably infinite number of elements. And if you are not quite familiar with these terms, that doesn’t matter. Just leave it at the intuitive thought that “well, that’s everything that you can somehow count.“

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Another important term is "event". What is an event?

Well, an event in a random experiment is each subset of the set of outcomes Ω. We thus absolutely need to define the set of outcomes in order to use the term.

We speak of an elementary event when we have a one-element subset of Ω.

The power set P(Ω), thus the set of all possible subsets of Ω, is called the event space of the random experiment. An event is accordingly always an element of P(Ω).

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Let’s look at an example: We roll a regular die. Clearly, you can determine Ω = {1, 2, 3, 4, 5, 6} as the set of outcomes.

Then “a 3 is rolled” is one of six possible elementary events.

It is the certain event that one of the numbers 1, 2, 3, 4, 5, or 6 will be rolled.

It’s impossible that none of the numbers 1, 2, 3, 4, 5, or 6 will be rolled. Consequently, this is an impossible event.

And again, we can freely define what interests us. “A 1 or 3 or 5 is rolled” is the “odd number” event and “a 2 or 3 or 5 is rolled” is the “prime number” event.

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This looks as follows in set notation:

T₁ = {3} is clear, T₂ = {1,2,3,4,5,6} is also not difficult. The impossible event is represented as the empty set T₃ = { } { }, the empty set is a subset of every set. Finally T₄ = {1,3,5} and T₅ = {2,3,5}.

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The essential terms are summarized again here.

The set of outcomes is the set Ω of all outcomes.

An event is each subset A of Ω.

All events make up the set of events and that is the power set P(Ω).

An elementary event is each one-element subset of Ω.

The certain event is A = Ω, and the impossible event A = { }.

For every event A there is a complementary event Ω \ A , thus the complement of A.

We can link the events with “and”, and then we have the intersection of the two events. And we can link the events with “or” and then we have the union of the two events. Too much set theory? Don’t worry, we’ll show examples next.

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Let’s roll a die again; that’s a nice, simple random experiment.

We assume that A₁ is the “odd number” event and A₂ is the “prime number” event.

The complementary event of A₁ is the “even number” event or the set of numbers 2, 4, and 6; the complementary event of A₂ is the “not a prime number” event, thus the set of numbers 1, 4, and 6.

If we link the two events with “and”, then we’re looking for numbers that are odd and prime numbers, which are clearly 3 and 5.

If we link the two events with “or”, then we’re looking for numbers that are either odd or a prime number, which are clearly the numbers 1, 2, 3, and 5.

When a die is rolled, it is certain that you’ll roll one of the numbers from 1 to 6.

It is impossible to roll a 7 … or an 8 or … whatever you want to insert here that differs from the whole numbers between 1 and 6.

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Let’s close with an example. If Real Madrid competes against Bayern Munich in soccer, there are three possible outcomes: Real Madrid wins, the teams end in a tie, or Bayern Munich wins.

Outcomes could be
“One team wins”, thus either Real Madrid wins or Bayern Munich wins.

Or
“There is no clear winner”, which means that the teams play to a tie.

Obviously, the two events are each the complementary event of the other.

Linking the two events with “and” doesn't work because the events rule each other out. You end up with the empty set, the impossible event, that neither team wins and the teams also do not end in a tie.

Linking with “or” is very easy. It leads to the set of outcomes Ω.

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That’s all for today. Many thanks for being here. I look forward to seeing you in the next episode.