Problem 3 - Arbeitsbuch Mathematik für Ingenieure

Question

Eine Münze mit den Seiten Wappen und Zahl werde dreimal geworfen. Betrachten Sie folgende Ereignisse: A: Gleiche Seite beim 1. und 2. Wurf \(B\) : Gleiche Seite beim 1. und 3. Wurf \(C:\) Gleiche Seite beim \(2 .\) und \(3 .\) Wurf Untersuchen Sie, ob unter der Laplace-Annahme die Ereignisse a) paarweise unabhängig b) vollständig unabhängig sind.

Step-by-Step Solution

Answer

Events A, B, and C are pairwise independent but not fully independent.

Step 1 – Define the Events

Define the outcomes for each event where 'W' indicates 'Wappen' and 'Z' indicates 'Zahl'. For event A, the first and second tosses are the same: {WWW, WWZ, ZZW, ZZZ}. For event B, the first and third tosses are the same: {WWW, WZW, ZWW, ZZW}. For event C, the second and third tosses are the same: {WWZ, WZZ, ZWZ, ZZZ}.

Step 2 – Calculate the Probability of Each Event

There are 8 possible outcomes when tossing the coin three times. So, the probability of each of these events can be calculated as follows: P(A) = 4/8 = 1/2 P(B) = 4/8 = 1/2 P(C) = 4/8 = 1/2

Step 3 – Calculate Joint Probabilities

Calculate the probability of the intersection of each pair of events: P(A ∩ B) = P( {WWW, ZZW} ) = 2/8 = 1/4 P(A ∩ C) = P( {WWZ, ZZZ} ) = 2/8 = 1/4 P(B ∩ C) = P( {WWW, ZZZ} ) = 2/8 = 1/4 Also, P(A ∩ B ∩ C) = P( {WWW, ZZZ} ) = 2/8 = 1/4

Step 4 – Check Pairwise Independence

To check pairwise independence, the probability of the intersection of two events must equal the product of their individual probabilities: P(A ∩ B) = 1/4, P(A) * P(B) = 1/2 * 1/2 = 1/4 (True) P(A ∩ C) = 1/4, P(A) * P(C) = 1/2 * 1/2 = 1/4 (True) P(B ∩ C) = 1/4, P(B) * P(C) = 1/2 * 1/2 = 1/4 (True) Thus, events A, B, and C are pairwise independent.

Step 5 – Check Complete Independence

To check complete independence, the probability of the intersection of three events must equal the product of their individual probabilities: P(A ∩ B ∩ C) = 1/4, P(A) * P(B) * P(C) = 1/2 * 1/2 * 1/2 = 1/8 (False) Thus, events A, B, and C are not fully independent.

Key Concepts

Probability Theory

Probability theory is the branch of mathematics that deals with the likelihood of different outcomes. It's used to quantify uncertainty. For instance, when tossing a coin, we expect either 'Wappen' (Heads) or 'Zahl' (Tails). Basic concepts in probability theory include:

• **Sample space**: All possible outcomes of a random experiment. In our coin example, it’s {W, Z}.
• **Event**: A subset of the sample space. For example, getting a head in a coin toss.
• **Probability**: The measure of how likely an event is to occur. Expressed as a number between 0 and 1.

We define probabilities by counting favorable outcomes and dividing by the total number of outcomes in the sample space. Tossing a coin three times has 8 possible outcomes (since it’s 2 outcomes each time and there are 3 tosses: 2^3 = 8). Probability theory allows us to model and analyze real-world situations to make informed predictions.

Pairwise Independence

Events are said to be pairwise independent if the occurrence of any two events does not influence each other. In simpler terms, whether one event happens doesn’t affect the probability of the other happening.
To check pairwise independence:

• Calculate the probability of each event individually. Using our example: P(A) = 1/2, P(B) = 1/2, P(C) = 1/2
• Calculate the joint probability of pairs: P(A ∩ B), P(A ∩ C), and P(B ∩ C).
• Ensure P(A ∩ B) = P(A) * P(B), P(A ∩ C) = P(A) * P(C), and P(B ∩ C) = P(B) * P(C). If all are true, the events are pairwise independent.

In the exercise:

• We found P(A ∩ B) = 1/4 and P(A) * P(B) = 1/4.
• P(A ∩ C) = 1/4 and P(A) * P(C) = 1/4.
• P(B ∩ C) = 1/4 and P(B) * P(C) = 1/4.

Since these conditions hold, events A, B, and C are pairwise independent.
Remember, works pairwise independence do **not** guarantee complete independence!

Complete Independence

Complete independence (or mutual independence) means that the occurrence of one event does not provide any information about the occurrence of any combination of the other events. It's a stronger condition than pairwise independence.
To check complete independence:

• Calculate the probability product of all events individually: P(A) * P(B) * P(C)
• Compare with the joint probability of all events: P(A ∩ B ∩ C)

Complete independence means P(A∩B∩C) should be the product of their individual probabilities. In our example:
* P(A) * P(B) * P(C) = 1/2 * 1/2 * 1/2 = 1/8 * P(A ∩ B ∩ C) = 1/4
Here, P(A) * P(B) * P(C) ≠ P(A ∩ B ∩ C), meaning A, B, and C are not completely independent even though they are pairwise independent. Complete independence is more stringent and requires every possible combination to be independent.

Laplace Assumption

The Laplace assumption is a fundamental concept in probability, also known as the principle of indifference. It states that if we have no reason to think otherwise, every outcome is equally likely.
In practical terms, this simplifies calculations and predictions. For our example:

• Each coin toss outcome (Wappen or Zahl) has an equal chance of 1/2.
• When dealing with multiple tosses, we assume all combinations of results (eight total) have an equal probability.

This assumption allows us to assign equal probabilities to each individual outcome before we know more about how they interact.
Thus, the probability of each outcome in our three-toss scenario is 1/8 (since each one is equally likely). From there, broader probabilities can be extracted (like P(A), P(B), and P(C)).
Without the Laplace assumption, calculating these probabilities would be much harder and could require more information. With it, students can focus on structure and principles, reducing computational complexity.