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Conditional probability

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

You pick a child from the class and see only: it is a girl. How likely is it that she has a bicycle? This is a conditional probability (die bedingte Wahrscheinlichkeit).
📄 Open worksheet (PDF, German)

The video

In the video we look only at the row or the column of the condition (die Bedingung) in the four-field table.

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

Contents of the video

  1. 0:00 Introduction
  2. 0:17 Only the girls
  3. 0:39 The other direction
  4. 1:08 The formula
  5. 1:32 Careful, mistake!
  6. 1:54 Try it yourself

The most important points

Conditional probability

The condition stands behind the bar: \(P(F \mid M)\) is read “P of F given M” (F = bicycle, M = girls). You look only at the row of the girls: \(12\) girls, \(5\) of them with a bicycle. \(P(F \mid M) = \frac{5}{12} \approx 42\,\%\).

Only the row of the girls counts.
Only the row of the girls counts.

The other direction

Condition bicycle: \(11\) children, \(5\) of them girls. \(P(M \mid F) = \frac{5}{11} \approx 45\,\%\). It holds: \(P(M \mid F) \neq P(F \mid M)\).

Now only the column with bicycle counts.
Now only the column with bicycle counts.

The formula

\(P(F \mid M) = \dfrac{P(M \text{ and } F)}{P(M)} = \dfrac{5/20}{12/20} = \dfrac{5}{12}\). Above: both features. Below: the condition.

Both features (green), condition (orange).
Both features (green), condition (orange).

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

A class has \(25\) children: \(13\) girls, \(10\) of them play an instrument; \(12\) boys, \(5\) of them play an instrument. How big is a) \(P(\text{instrument} \mid \text{boy})\) b) \(P(\text{girl} \mid \text{instrument})\)?

Show solution

a) \(\frac{5}{12}\) b) \(15\) children play an instrument, \(10\) of them are girls: \(\frac{10}{15} = \frac{2}{3}\)

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: Bedingte Wahrscheinlichkeit (Vierfeldertafel, Zeile oder Spalte der Bedingung). 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.