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Law of large numbers

This page was translated from German with AI. It has not been checked by a native speaker. Original page in German

With few trials, chance plays a big role. With very many trials, the relative frequency (die relative Häufigkeit) is close to the probability.
📄 Open worksheet (PDF, German)

The video

In the video we roll a dice up to \(1000\) times and estimate the probability for a drawing pin.

Subtitles are available in ten languages. Choose them at the bottom right of the video.

Contents of the video

  1. 0:00 Introduction
  2. 0:20 More rolls
  3. 0:38 The law of large numbers
  4. 0:58 Estimating a probability
  5. 1:21 Careful, mistake!
  6. 1:42 Try it yourself

The most important points

\(10\) rolls, \(4\) sixes: relative frequency \(\frac{4}{10} = 0{,}4\). But the probability is only \(\frac{1}{6} \approx 0{,}17\). The dice is not broken: with few rolls, chance plays a part.

Sixes after 10, 50, 100, 500 and 1000 rolls.
Sixes after \(10\), \(50\), \(100\), \(500\) and \(1000\) rolls.

Law of large numbers

The more often you repeat an experiment, the closer the relative frequency is to the probability (das Gesetz der großen Zahlen).

The curve comes closer to 1/6.
The curve comes closer to \(\frac{1}{6}\).

If you do not know the probability, you estimate it: a drawing pin thrown \(1000\) times, \(620\) times on its back: \(P \approx 0{,}62\).

Back or side?
Back or side?

Careful: in \(600\) rolls there are not exactly \(100\) sixes. The exact number is chance.

The worksheet

Here you can see the worksheet for the video (in German). First it explains everything again, then there are practice tasks at three levels. You can also open it as a PDF and print it.

📄 Open worksheet (PDF, German)

The worksheet – Page 1
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The worksheet – Page 2
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The worksheet – Page 3
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The worksheet – Page 4
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Show the worksheet solutions
The worksheet – Page 5
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Try it yourself

Task

A coin is tossed \(200\) times, heads comes up \(112\) times. What is the relative frequency? Does it fit the probability \(0{,}5\)?

Show solution

\(\frac{112}{200} = 0{,}56\). That is close to \(0{,}5\).

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: Gesetz der großen Zahlen (relative Häufigkeit nähert sich der Wahrscheinlichkeit, Wahrscheinlichkeit schätzen). Stelle mir nacheinander fünf kurze Aufgaben. Warte immer auf meine Antwort. Sag mir dann, ob sie richtig ist, und erkläre kurz, warum. Benutze einfache deutsche Sätze.

How this was made: The idea, structure and examples of the series are mine. I made them with the help of AI: a language model (LLM) built the slides of the video following my instructions. The voice was generated with Qwen3-TTS, modelled on my own voice.

The subtitles, the translations, the worksheet and this page were also made with the help of AI. The translations have not been checked by native speakers.

The videos, worksheets, texts and prompts in this series are licensed under CC BY 4.0: you may use, change and share them if you name the source.