# How can I use statistics and probabilities

### Friendship saved by the relationship between relative frequency and probability

Paparim and Jonas often play Mensch anger dich nicht together.

Paparim distrusts Jonas. Somehow he has a 6 more often. That's why Jonas wins more often.

Paparim takes the dice home and plans to roll 100 times.

After rolling the dice 100 times, he has the following result:

number rolled | 1 | 2 | 3 | 4 | 5 | 6 |

number | 10 | 15 | 17 | 16 | 16 | 26 |

In fact, the 6 appears more often as a result. It appears 26 times or, calculated on the basis of 100 rolls, with a relative frequency of 26%.

The relative frequency of a result is the number of results divided by the number of all attempts: $$ text {relative frequency} = frac {text {number of results}} {text {number of attempts}} $$

Image: Druwe & Polastri

### What goes forward also goes backwards or security with math

Paparim is in a very bad mood. Should his friend have tricked him so much ...

The next day, Paparim confronts Jonas with his result. Jonas says it was a coincidence that the 6 came 26 times. So he rolls the dice again, but now 10,000 times. In the end, Jonas and Paparim state:

Eye count | 1 | 2 | 3 | 4 | 5 | 6 |

Number of litters | 1666 | 1665 | 1665 | 1667 | 1667 | 1670 |

In fact, the 6 was still thrown more often. But their relative frequency is only 16.7%. Apparently the dice is fair after all. Each digit occurs with a probability of around 16.7%. The relative frequency of this experiment is getting closer and closer to the probability.

The relative frequency of a result approximates the probability of the result if the experiment is carried out frequently.

Image: fotolia.com (p! Xel 66)

### What goes forward also goes backwards or security with math

The probability of being struck by lightning in Germany is $$ frac 1 {6,000,000} $$.

What does this mean for the relative frequency? The relative frequency is just as great as the probability.

In absolute numbers this means for Germany:

Of 81,000,000 Germans annually about

$$ 81,000,000 cdot 1 / (6,000,000) = 13.5 $$

People struck by lightning.

If a random experiment is carried out frequently, the probability corresponds to the relative frequency. Multiplying the relative frequency by the number of attempts gives an expected absolute frequency. Mathematically, this value is called the expected value.

Image: adpic picture agency (A. Buß)

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### Safe to the cinema with mathematics, thanks to the probability.

Louisa and Fabienne have a bag full of yellow and red marbles. There are a total of 10 balls in the bag. You want to annoy Peter a little because Peter likes Louisa so much. You give him the following riddle:

Fabienne says to Peter: “When you find out how many balls are yellow, red and blue, Louisa invites you to the cinema. You can draw a ball 100 times and pack it back again. There are ten balls in total. “Peter is happy. He draws and receives the following result:

yellow marbles | red marbles | blue marbles |
---|---|---|

19 | 81 | 0 |

So Peter says for sure: There are only 2 yellow and 8 red balls in the bag. The movie night was very nice. :-)

A little more math:

relative frequency of yellow marbles: $$ 19/100 $$

Probability of yellow marbles: $$ 0.2 $$

Relative frequency of red marbles: $$ 81/100 $$

Probability of red marbles: $$ 0.8 $$

If you know the total number of balls, the relative frequency multiplied by the total number gives the approximate number of balls.

$$ 19/100 * 10 = 19/10 approx 2 $$ yellow balls

This also works with probability. The probability multiplied by the total number gives the number of balls.

$$ 0.2 * 10 = 2 $$ yellow balls

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