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Stochastic independence

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

Two events are independent (unabhängig) if one does not change the probability of the other.
📄 Open worksheet (PDF, German)

The video

In the video we check with two-way tables: are glasses and girls connected? And bikes and girls?

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

Contents of the video

  1. 0:00 Introduction
  2. 0:18 Comparing probabilities
  3. 0:42 Dependent
  4. 1:04 The formula
  5. 1:36 Careful, mistake!
  6. 1:57 Try it yourself

The most important points

Independent

\(A\) and \(B\) are independent if \(P(B \mid A) = P(B)\). Otherwise they are dependent (abhängig).

P(B) = 12/40 = 30\,\% and P(B \mid M) = 6/20 = 30\,\%: independent.
\(P(B) = \frac{12}{40} = 30\,\%\) and \(P(B \mid M) = \frac{6}{20} = 30\,\%\): independent.

Bike: \(P(F) = \frac{11}{20} = 55\,\%\), but \(P(F \mid M) = \frac{5}{12} \approx 42\,\%\): dependent.

Different probabilities: dependent.
Different probabilities: dependent.

The formula

\(A\) and \(B\) are independent if \(P(A \cap B) = P(A) \cdot P(B)\). Glasses: \(\frac{6}{40} = 0{,}15 = 0{,}5 \cdot 0{,}3\).

The formula checked with the two-way table.
The formula checked with the two-way table.

Careful: after five heads, tails is not more likely. The tosses are independent, \(P(\text{tails}) = \frac{1}{2}\).

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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Show the worksheet solutions
The worksheet – Page 4
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Try it yourself

Task

\(50\) children: \(30\) girls, \(12\) of them with a dog; \(20\) boys, \(8\) of them with a dog. Are „dog“ and „girl“ independent?

Show solution

\(P(H) = \frac{20}{50} = 40\,\%\), \(P(H \mid M) = \frac{12}{30} = 40\,\%\): independent.

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: stochastische Unabhängigkeit (P(B | A) = P(B), P(A ∩ B) = P(A) · P(B), Vierfeldertafel). 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.