Baumdiagramm: Ziehen mit Zurücklegen

In diesem Artikel erkläre ich dir zunächst was ein Baumdiagramm ist, dann wie und warum man es in der Wahrscheinlichkeitsrechnung verwendet und anschließend, wie ein Baumdiagramm für den Fall „Ziehen mit Zurücklegen“ erstellt wird. Als letztes erkläre ich dir die beiden wesentlichen Rechenregeln an einem Baumdiagramm.

Was du vorher wissen solltest:

Was ist ein Baumdiagramm?

Baumdiagramm

Baumdiagramme in der Wahrscheinlichkeitsrechnung:

Sie werden zur Berechnung von Wahrscheinlichkeiten von Ereignissen bei mehrstufigen Zufallsexperimenten eingesetzt, da sie dabei helfen einen guten Überblick zu behalten. Ein Baumdiagramm, welches dann mit entsprechenden Wahrscheinlichkeiten ergänzt wird, nennt man dementsprechend in der Wahrscheinlichkeitsrechnung auch Wahrscheinlichkeitsbaum. Diese Wahrscheinlichkeiten werden in der Regel als Bruch oder Dezimalzahl angegeben.

Was bedeutet: „Ziehen mit Zurücklegen“?

Wie ich schon erwähnt habe, werde ich dir heute in diesem Artikel das Baumdiagramm für „Ziehen mit Zurücklegen“ erklären. Dazu schauen wir uns zunächst an, was „Ziehen mit Zurücklegen“ bedeutet.

Urne: Ziehen mit Zurücklegen

Hierfür habe ich eine Urne vorbereitet. In dieser Urne sind, wie du siehst, rote und blaue Kugeln. Die relative Häufigkeit für eine rote Kugel ist somit [latex size=25]\frac {3}{5}[/latex] und für eine blaue [latex size=25]\frac {2}{5}[/latex].

 P(„rot”) = [latex size=30]\frac {3}{5}[/latex]

P(„blau”) = [latex size=30]\frac {2}{5}[/latex]

Jede Kugel, die wir nun aus dieser Urne ziehen werden, legen wir wieder zurück, sodass nach jedem Zug die Ausgangssituation hergestellt wird. Das heißt also, dass sich die Wahrscheinlichkeiten von den Ausgängen im Gegensatz zu „Ziehen ohne Zurücklegen“ nie, also auch nicht beim zweiten oder dritten Zug, ändern, da zu jedem Zieh-Zeitpunkt 3 von 5 Kugeln rot und 2 von 5 blau sind.

Erstellung eines Baumdiagramms:

Die Erstellung eines Baumdiagramms möchte ich dir nun anhand dieser Urne erklären. Aus dieser Urne ziehen wir nun eine Kugel, legen sie zurück in die Urne und ziehen dann eine zweite Kugel, also für „Ziehen mit Zurücklegen“:

1. Als erstes überlegen wir uns wieviele verschiedene Möglichkeiten dieser Zug hat! In diesem Fall sicherlich zwei, denn wir können eine rote oder eine blaue Kugel ziehen. Das heißt, dass wir nun zwei Abzweigungen brauchen (allgemein: eben genau gleich viele Abzweigungen wie Möglichkeiten).

Baumdiagramm Erstellung

Am Anfang dieser Abzweigung schreibt man gerne die Ausgangssituation hin. Hier steht r für rot und b für blau.

2. Nachdem wir nun die Anzahl der Abzweigungen ermittelt haben, werden die Enden dementsprechend beschriftet. Eine Abzweigung steht für den Ausgang rot, die Andere für blau.

Baumdiagramm Erstellung

Alternativ zu zwei farbigen Punkten, kannst du bei dieser Situation auch wieder gerne mit einem r und einem b beschriften.

3. Nun werden die relativen Häufigkeiten an die Seite der jeweiligen Äste hingeschrieben.

Baumdiagramm Erstellung

In diesem Fall hat die rote Kugel die relative Häufigkeit [latex size=25]\frac {3}{5}[/latex], da drei von fünf Kugeln rot sind und die blaue Kugel [latex size=25]\frac {2}{5}[/latex], da zwei von fünf Kugeln blau sind.

Die erste von zwei Ziehungen ist nun beendet! Nun starten wir mit der zweiten Ziehung und beginnen diese drei Schritte von vorne!

Wir überlegen uns wieder wieviele Möglichkeiten wir nun haben, denn das entspricht ja der Anzahl an Ästen. Wenn wir vorher eine rote oder eine blaue Kugel gezogen haben (diese haben wir wieder zurückgelegt), haben wir wieder jeweils zwei Möglichkeiten.

Baumdiagramm Erstellung

Auch hier beschriften wir nun die Enden. Wir haben wieder jeweils die Möglichkeit rot und blau zu ziehen.

Baumdiagramm Erstellung

Als letztes beschriften wir wieder die Äste mit den relativen Häufigkeiten. Hier überlegen wir uns, dass die rote Kugel immernoch die Wahrscheinlichkeit [latex size=25]\frac {3}{5}[/latex] hat, da wir die vorherige Kugel, egal ob rot oder blau, wieder in die Urne zurückgelegt haben und somit sind wieder, wie am Anfang, 3 von 5 Kugeln rot. Genauso ist die Wahrscheinlichkeit von einer blauen Kugel immernoch [latex size=25]\frac {2}{5}[/latex].

Baumdiagramm Erstellung

Dies ist das vollständig ausgefüllte Baumdiagramm. Würden wir noch eine dritte Kugel ziehen, dann würden wir diese drei Schritte wieder wiederholen.

Die Rechenregeln an einem Baumdiagramm:

An einem Baumdiagramm gibt es grundsätzlich zwei Rechenregeln:

Pfadmultiplikation

Um die Endwahrscheinlichkeiten zu bestimmen, wird entlang eines Astes multipliziert. Die Endwahrscheinlichkeiten stehen jeweils für die Wahrscheinlichkeit eines bestimmten Versuchsausgangs.

Summenregel

Verschiedene Endwahrscheinlichkeiten werden addiert. Diese Regel wird dann gebraucht, wenn ein Ereignis mehrere Versuchsausgänge beinhaltet.

Pfadmultiplikation und Summenregel

Beispiele:

Beispiel 1: Gesucht ist die Wahrscheinlichkeit von zwei roten Kugeln:

Pfadmultiplikation

Wie du siehst handelt es sich um einen einzigen Ast von dem wir nun die Endwahrscheinlichkeit suchen. Hierzu verwenden wir die Produktregel (sie wird oft auch Pfadmultiplikationsregel genannt):

P(r,r) = P(Summenregel,) = [latex size=30]\frac {3}{5}[/latex] x [latex size=30]\frac {3}{5}[/latex] = [latex size=30]\frac {9}{25}[/latex]

Beispiel 2: Gesucht ist die Wahrscheinlichkeit von einer blauen Kugel:

Summenregel

Wie du siehst handelt es sich um zwei verschiedene Äste von denen wir nun die Endwahrscheinlichkeiten jeweils mit der Produktregel berechnen und diese dann mithilfe der Summenregel addieren. Die Formulierung „eine blaue Kugel“ sagt ja keinesfalls aus, dass diese Kugel als erstes gezogen werden muss. Diese blaue Kugel kann offensichtlich als erstes oder als zweites gezogen werden, sodass es genau diese beiden Äste sind, von denen wir die Wahrscheinlichkeit ermitteln müssen:

P(r,b) = P(,) = [latex size=30]\frac {3}{5}[/latex] x [latex size=30]\frac {2}{5}[/latex] = [latex size=30]\frac {6}{25}[/latex]

P(b,r) = P(,) = [latex size=30]\frac {2}{5}[/latex] x [latex size=30]\frac {3}{5}[/latex] = [latex size=30]\frac {6}{25}[/latex]

P(,) + P(,) = [latex size=30]\frac {6}{25}[/latex] + [latex size=30]\frac {6}{25}[/latex] = [latex size=30]\frac {12}{25}[/latex]

 

Tipps zur Erstellung:

1. Als erstes solltest du dir darüber im klaren sein, wie viele Abzweigungen du an jeder Stufe brauchst. Hierzu ist es sinnvoll sich die Frage zu stellen wofür die Abzweigungen in der Aufgabenstellung stehen (oft sind es die Farben der Kugeln, die verschiedenen Farben des Glücksrades, das Geschlecht,…). Tipps Baumdiagramm

2. Nun schaue bitte wieviele Stufen dein Baumdiagramm haben muss. Stelle dir also die Frage, wofür deine Stufen stehen (oft ist es die Anzahl an Handlungen, also wie oft eine Kugel entnommen wird, wie oft ein Rad gedreht wird, wieviele Personen getestet werden usw.).

Tipps Baumdiagramm

3. Du kannst deine Wahrscheinlichkeiten an einem Ast überprüfen, denn die Wahrscheinlichkeiten an jeder Abzweigung müssen addiert 1 ergeben.

Tipps Baumdiagramm

4. Auch deine Endwahrscheinlichkeiten lassen sich leicht überprüfen, denn alle Endwahrscheinlichkeiten müssen ebenfalls addiert 1 ergeben.

 

Auf YouTube ansehen: >>>Hier klicken<<<

 

 

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