# What is 4 out of 25

### Don't get angry

Tony chatted a few more friends and she

start a new game of "Mensch ärgere Dich nicht".

So everyone is waiting for a 6 so that they can place a pawn on the field.

Wild methods do the rounds: roll the dice with the left, take a dice cup, cast spells, ...

But now let's be sober: How big is it **chance**to roll a 6?

The die has the six numbers 1, 2, 3, 4, 5, 6. You want a 6. You can also say: The 6 is that **cheap** Result.

The 6 is one of the six numbers. That sounds like a share! 1 out of 6 is cheap. As a fraction: $$ 1/6 $$.

Mathematicians say: The **probability**to roll a 6 is $$ 1/6 $$.

Image: Michael Fabian

### And the relative frequency?

How does the probability fit with these

Frequencies together, you might ask yourself.

Why did you draw these tally sheets and calculate relative frequencies when rolling the dice ...

**example**: Roll the dice 60 times

Eye count | Absolute frequency | Relative frequency |
---|---|---|

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If you really roll the dice, the portion of the 6's is almost never exactly $$ 1/6 $$. The more often the cube experiment is carried out

(1000 times, 10,000 times ...), the closer the proportion of 6s to $$ 1/6 $$.

But it's somehow logical: A dice has 6 identical sides, what else can happen than that you roll every number with the portion of $$ 1/6 $$. That's the point! You expect $$ 1/6 $$. That's what mathematicians call **probability**.

The **probability** of a result is the **expected relative frequency** this result.

At a **Random experiment** you can't predict the outcome.

- Throw the dice
- Toss a coin
- Throw Lego bricks
- Pull loose
- Spin the wheel of fortune

Calculation of the relative frequency: $$ relative \ frequency = frac {ab solute \ frequency} {total number} $$

You can find relative frequencies both in **Fractions, decimal fractions** as well as in **Percent (%)** specify.

Example: $$ frac {1} {4} = frac {25} {100} = 0.25 = 25% $$

### Examples of probabilities

The probability has the symbol $$ p $$. That comes from the English: probability.

**Wheel of fortune**

Result set: {RED; BLUE; YELLOW}

Probability for ROT: $$ p = 2/6 = 1/3 $$

Probability for BLUE: $$ p = 1/6 $$

Probability for YELLOW: $$ p = 3/6 = 1/2 $$

**urn**

Result set: {1; 2; 3; 4}

Probability for 1: $$ p = 3/8 $$

Probability for 2: $$ p = 2/8 = 1/4 $$

Probability for 3: $$ p = 2/8 = 1/4 $$

Probability for 4: $$ p = 1/8 $$

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### Immediately likely

Random experiments in which all results are equally likely are easy to calculate. When rolling the dice, all numbers from 1 to 6 have the same probability $$ p = 1/6 $$.

Further examples:

**Toss a coin**

Result set: {head; Number}

number of **possible** Results: 2

Probability for one **cheap** Result: $$ p = frac {1} {2} $$

**Card game**

Result set: {cross 7; Cross 8; ..., King of Diamonds; Ace of diamonds}

number of **possible** Results: 32

Probability for one **cheap** Result: $$ p = frac {1} {32} $$

What is the probability of drawing a cross card?**solution**:

Number of possible outcomes: 32

Number of favorable results: 8

The probability of drawing a cross card is $$ p = frac {8} {32} = frac {1} {4} = 0.25 $$.

If in a random experiment all possible outcomes occur with the same probability, you calculate the probability $$ p $$ as follows:

$$ p = frac {number \ of \ favorable \ results} {number \ of \ possible \ results} $$

### General information on probability

The probability is a proportion. That is, it lies **between 0 and 1**. And what about 0 and 1?

**Example dice**:

Result set: {1; 2; 3; 4; 5; 6}

Impossible event:

- Event "number greater than 6": {}
- $$ p = 0 $$

Possible event:

- Event "even number": {2; 4; 6}
- $$ p = 3/6 = 1/2 $$

Safe event:

- Event "Number less than 7 but greater than 0": {1; 2; 3; 4; 5; 6}
- $$ p = 1 $$

The following applies to the probability $$ p $$:

- $$ p = 0 $$: The event never occurs, the event is
**impossible**. - $$ 0 lt p lt 1 $$: The event is
**possible**. - $$ p = 1 $$: The event always occurs. The event is
**for sure**.

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