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Tree diagram and path rules

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

An urn contains \(2\) red and \(3\) blue balls. You draw twice and put the ball back. With a tree diagram (das Baumdiagramm) you find all possibilities and their probabilities.
📄 Open worksheet (PDF, German)

The video

In the video the tree grows branch by branch; then we calculate along the paths.

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 The tree diagram
  3. 0:50 The product rule
  4. 1:25 The sum rule
  5. 1:49 Careful, mistake!
  6. 2:10 Try it yourself

The most important points

Tree diagram

Every branch is an outcome with its probability: red \(\frac{2}{5}\), blue \(\frac{3}{5}\). The branches of one node add up to \(1\).

Four paths (die Pfade): red, red · red, blue · blue, red · blue, blue.
Four paths (die Pfade): red, red · red, blue · blue, red · blue, blue.

The two path rules

Multiply along a path: \(\frac{2}{5} \cdot \frac{2}{5} = \frac{4}{25}\). Add between paths: exactly one red ball \(= \frac{6}{25} + \frac{6}{25} = \frac{12}{25}\).

4/25 + 6/25 + 6/25 + 9/25 = 1
\(\frac{4}{25} + \frac{6}{25} + \frac{6}{25} + \frac{9}{25} = 1\)
The two green paths fit the event “exactly one red ball”.
The two green paths fit the event “exactly one red ball”.

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
Page 2/4
The worksheet – Page 3
Page 3/4
Show the worksheet solutions
The worksheet – Page 4
Page 4/4

Try it yourself

Task

You toss a coin twice. How big is the probability of a) two heads b) exactly one head?

Show solution

a) \(\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}\) b) \(\frac{1}{4} + \frac{1}{4} = \frac{1}{2}\)

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: Baumdiagramm und Pfadregeln (Produktregel, Summenregel). 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.