# How do we use the probability

## Basic concepts of probability theory

### Probability Theory - Examples

In the learning video we already have a little insight into the** probability calculation** given. In the following you will find two further examples of probabilities when rolling the dice, which we will calculate for you in writing:

### example 1

Event: How likely is it to roll a $ 3 $?

Event set: $ E = \ {3 \} $

Probability: $ P (E) = P (3) = \ frac {1} {6} $

### Example 2

An event can also consist of several outcomes: How likely is it to roll a $ 2 or a $ 3?

Event set: $ E = \ {2; 3 \} $

Probability: $ P (E) = \ frac {2} {6} = \ frac {1} {3} $

Of course, there is also the possibility that the result set and the event set match. In our example this would be the event: How likely is it to roll a 1, 2, 3, 4, 5 or 6? Since this event includes all outcomes that can occur, it is also called that **safe event**.

- The result set contains all results that can occur.

- An event consists of elements of the result set. That is, an event is a subset of the result set.

- When an event includes all elements of the result set, the event is called a safe event. Result set and event set are then identical.

### Dealing with probabilities

Knowledge of percentages is very important for a safe handling of probabilities in math. You should understand the relationship between decimals and percentages and be able to convert fractions into decimals and percentages, and vice versa.

The probability of throwing a certain number when throwing it once is the same for every number and is calculated using the relative proportion (the relative frequency):

$ relative \ part = \ frac {part (s)} {whole} = \ frac {1} {6} ~ \ approx ~ 0.17 ~ \ widehat {=} ~ 16 ~ \% $

If you do not feel fit in this area yet, you should look again at our learning text on this topic.

### Counter event

A counter-event contains all results that do not count towards the event. Counter-event and event taken together are the same as the result set, namely all results that can occur at all. Much more important than this connection, however, is that the sum of the probabilities of the event and counter-event must always add up to 1 or 100%.

The probability of rolling a 1 is $ \ frac {1} {6} $.

The counter-event $ \ overline {E} $ now covers all other results that do not belong to the event $ E $; So, rolling a 2, 3, 4, 5 or 6.

Since the sum of event and counter-event must always add up to 1, we can calculate the probability of the counter-event:

$ P (\ overline {E}) = 1 - P (E) = 1 - \ frac {1} {6} = \ frac {5} {6} \ approx 0.83 ~ \ widehat {=} ~ 83 ~ \% $

In the **Exercises** you can now test your knowledge of probability theory. Good luck with it!

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