Erwartungswert ohne Einsatz | Erklärung und Beispiel

In diesem Artikel erkläre ich dir wie man den Erwartungswert ohne Einsatz bestimmt. Hierzu habe ich ein anschauliches Beispiel vorbereitet, an dem ich das Vorgehen bei der Bestimmung erkläre. Danach erkläre ich noch die mathematische Formel.

Was du vorher wissen solltest:

Was ist der Erwartungswert?

Erklärung Erwartungswert ohne Einsatz an einem Beispiel:

Situation:

Dieses Glücksrad wird gedreht und je nachdem wo es stehen bleibt erhält der Spieler einen Gewinn.

Wenn das Glücksrad auf Rot stehen bleibt geht der Spieler allerdings leer aus und erhält nichts. Die Zufallsvariable X gibt den Gewinn des Spielers in Euro an.

Frage: Wieviel Euro Gewinn kann der Spieler durchschnittlich auf lange Sicht pro Spiel erwarten?

Lösung: Für den Erwartungswert brauchen wir zunächst die Wahrscheinlichkeitsverteilung!

Zunächst legen wir also eine korrekt beschriftete Tabelle an. Die erste Zeile wird mit den möglichen Ausgängen der Zufallsvariable beschriftet (Erklärung: Wahrscheinlichkeitsverteilung)! Hier sind es die möglichen Gewinne, die der Spieler erhalten kann.

Nun ermitteln wir die zugehörigen Wahrscheinlichkeiten und tragen diese in die Tabelle ein:

Die Farbe rot hat die Wahrscheinlichkeit 50%, als Dezimalzahl also 0,5. Wenn des Glücksrad auf rot stehen bleibt, erhält der Spieler nichts, also trage ich die 0,5 unter die Null ein. Geld und Grün haben beide die Wahrscheinlichkeit 25%, als Dezimalzahl dann 0,25 und diese Zahlen trage ich nun unter die 1 und die 5 ein.

Das ist die vollständig ausgefüllte Wahrscheinlichkeitsverteilung mit der wir nun den Erwartungswert berechnen.

Die Berechnung ist nun nicht mehr schwer, denn die möglichen Merkmale der Zufallsvariable werden mit den zugehörigen Wahrscheinlichkeiten multipliziert und diese dann addiert.

Umgangssprachlich würde man es wohl so formulieren:

„Oben mal unten + oben mal unten +…“ bis man auch die letzte Spalte verechnet hat.

Der Erwartungswert wird häufig mit dem griechischen Buchstaben µ oder mit E(x) gekennzeichnet!

Die Rechnung lautet also:

µ = 0 • 0,5 +1 • 0,25 + 5 • 0,25 = 1,5

Der Erwartungswert ist bei diesem Beispiel also 1,5. Da der Erwartungswert immer die selbe Einheit wie die Wahrscheinlichkeitsverteilung hat, steht in diesem Sachkontext die 1,5 für 1,50€.

Antwort: Der Spieler kann auf lange Sicht durchschnittlich einem Gewinn von 1,50€ erwarten.

Mathematische Definition von Erwartungswert:

Die mathematische Formel sieht zunächst komplizierter aus als sie eigentlich ist, denn X1 bis Xn sind nichts anderes als die möglichen Ausgprägungen der Zufallsvariable. Diese standen oben in der Wahrscheinlichkeitsverteilung und werden mit den zugehörigen Wahrscheinlichkeiten multipliziert. Die erste Ausprägung X1 also mit P(X = X1), die zweite dann mit P(X = X2), bis man schließlich bei der letzten Ausprägung (diese wird mit Xn bezeichnet) angekommen ist.

Diese Formel beschreibt also genau das Vorgehen, welches ich dir am Beispiel erklärt habe!

Hilfreiche Tipps:

1. In der Aufgabenstellung habe ich folgende Formulierung gewählt:

„Wieviel Euro Gewinn kann der Spieler durchschnittlich auf lange Sicht pro Spiel erwarten?“

Die Wörter „durchschnittlich“ „auf lange Sicht“ und „pro …“ sind Signalwörter, die dir sofort sagen sollten, dass du nun die Aufgabe hast den Erwartungswert zu berechnen!

2. Da der Erwartungswert häufig in dem Sachkontext gedeutet werden soll, bietet es sich auch an genau diese Signalwörter zu verwenden!

3. Der Erwartungswert hat immer die selbe Einheit wir die Wahrscheinlichkeitsverteilung bzw. wie die Zufallsgröße

 

Video auf Youtube anschauen: >>>Hier klicken<<<

 

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