Problem 3 - Arbeitsbuch Mathematik für Ingenieure

Question

Ein zufällig ausgewählter Hörer einer bestimmten Vorlesung besteht die Semestralklausur mit Wahrscheinlichkeit \(0.75 .\) Fällt er durch die Klausur, dann nimmt er an einer mündlichen Nachprüfung teil. Die Gesamtprüfung gilt als bestanden, wenn die Klausur oder die Nachprüfung bestanden wurden. Die Wahrscheinlichkeit, dass ein in der Klausur durchgefallener Student dem Prüfer \(A\) bzw. \(B\) bzw. \(C\) zugeteilt wird, betrage \(0.5\) bzw. \(0.3\) bzw. \(0.2\). Die Wahrscheinlichkeit, bei Prüfer \(A\) bzw. \(B\) bzw. \(C\) die mündliche Nachprüfung zu bestehen, betrage \(0.6\) bzw. \(0.3\) bzw. \(0.8\). a) Zeichnen Sie ein Pfaddiagramm, das obige Situation beschreibt, und beschriften Sie Äste und Knoten.Ein zufällig ausgewählter Hörer einer bestimmten Vorlesung besteht die Semestralklausur mit Wahrscheinlichkeit \(0.75 .\) Fällt er durch die Klausur, dann nimmt er an einer mündlichen Nachprüfung teil. Die Gesamtprüfung gilt als bestanden, wenn die Klausur oder die Nachprüfung bestanden wurden. Die Wahrscheinlichkeit, dass ein in der Klausur durchgefallener Student dem Prüfer \(A\) bzw. \(B\) bzw. \(C\) zugeteilt wird, betrage \(0.5\) bzw. \(0.3\) bzw. \(0.2\). Die Wahrscheinlichkeit, bei Prüfer \(A\) bzw. \(B\) bzw. \(C\) die mündliche Nachprüfung zu bestehen, betrage \(0.6\) bzw. \(0.3\) bzw. \(0.8\). a) Zeichnen Sie ein Pfaddiagramm, das obige Situation beschreibt, und beschriften Sie Äste und Knoten.

Step-by-Step Solution

Answer

Draw a tree starting from the initial pass/fail node, then branch out for examiners and their corresponding pass/fail probabilities.

Step 1 - Understand the Problem

The problem involves a student who takes an exam with a given probability of passing. If the student fails, they take an oral retake with one of three examiners. We need to draw a tree diagram to represent this scenario and label each branch with the appropriate probabilities.

Step 2 - Initial Probability

Start by noting the probability that a student passes the initial exam and the probability that they fail: Pass (P) with probability 0.75, Fail (F) with probability 0.25.

Step 3 - Conditional Probabilities for the Oral Retake

If a student fails the initial exam, they are assigned to one of three examiners: Examiner A (E_A) with probability 0.5, Examiner B (E_B) with probability 0.3, Examiner C (E_C) with probability 0.2.

Step 4 - Probabilities of Passing the Oral Exam

Each examiner has a different probability of passing the student in the oral exam: Pass with Examiner A (P_A) is 0.6, Pass with Examiner B (P_B) is 0.3, Pass with Examiner C (P_C) is 0.8.

Step 5 - Create Tree Diagram

Draw the tree diagram starting from the initial node (start). From the start node, draw two branches: one for passing the initial exam (with probability 0.75) and one for failing the initial exam (with probability 0.25). From the fail node, draw three more branches representing the assignment to examiners A, B, and C (with respective probabilities 0.5, 0.3, and 0.2). Finally, from each of these nodes, draw branches for passing and failing the oral exam with the respective probabilities for each examiner.

Step 6 - Label Branches and Nodes

Label each branch with the corresponding probability. For example, the branch representing passing the initial exam would be labeled with 0.75; the branch representing failing and then being assigned to Examiner A would be labeled with 0.25 * 0.5, and so on. Ensure each node clearly shows whether it represents passing or failing at each stage.

Key Concepts

Tree Diagrams

A tree diagram is a graphical way to represent all possible outcomes of a given scenario. This helps visually break down complex probabilities into manageable parts.
In our example, a student can either pass or fail an initial exam. If the student fails, they are assigned to one of three examiners for an oral retake. Each examiner has different probabilities of passing the student.
To start:

• Draw a starting point.
• Branch out to show the outcomes of passing or failing the exam.
• If the student fails, create branches for being assigned to Examiners A, B, and C with their specific probabilities (0.5, 0.3, 0.2).
• From each examiner’s node, add branches for passing or failing the oral retake with their specific probabilities (e.g., Examiner A’s pass rate is 0.6).

This structured representation helps you calculate the combined probabilities of all outcomes by following paths from the start to the finish.

Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has already occurred. It is denoted as \( P(A|B) \), which reads as the probability of A given B.
In our exercise, we need to find the conditional probabilities of a student passing the oral retake given they failed the initial exam. If the student is assigned to one of three examiners (A, B, or C) after failing, we calculate the conditional probability of passing with each examiner.
We know:\( P(\text{Pass Initial Exam}) = 0.75 \) and \( P(\text{Fail Initial Exam}) = 0.25 \). If they fail, they have the following probabilities for being assigned to examiners:

• Examiner A: \( P(E_A | \text{Fail}) = 0.5 \)
• Examiner B: \( P(E_B | \text{Fail}) = 0.3 \)
• Examiner C: \( P(E_C | \text{Fail}) = 0.2 \)

For each examiner, the probability of passing the oral retake is:

• Examiner A: \( P(\text{Pass} | E_A) = 0.6 \)
• Examiner B: \( P(\text{Pass} | E_B) = 0.3 \)
• Examiner C: \( P(\text{Pass} | E_C) = 0.8 \)

By multiplying these conditional probabilities along with the initial failure rate, we get the total probability of passing after failing the initial exam.

Examination Statistics

Examination statistics often involve analyzing probabilities to determine a student's likelihood of success. In our problem, we are interested in the overall probability that a student passes either the initial exam or the oral retake.
We start with the given probabilities:

• Probability of passing the initial exam: 0.75
• Probability of failing the initial exam: 0.25

If a student fails, they have another chance through the oral retake. The probability of passing the oral exam depends on the examiner:

• For Examiner A: \( P(\text{Pass Oral}|E_A) = 0.5 \times 0.6 = 0.3 \)
• For Examiner B: \( P(\text{Pass Oral}|E_B) = 0.3 \times 0.3 = 0.09 \)
• For Examiner C: \( P(\text{Pass Oral}|E_C) = 0.2 \times 0.8 = 0.16 \)

The total probability of passing by taking the oral exam if the student fails the initial exam is the sum of these probabilities: \( 0.3 + 0.09 + 0.16 = 0.55 \).
So, the overall probability of passing the module (considering both the initial exam and the oral exam) is:
\[ P(\text{Pass Overall}) = 0.75 + (0.25 \times 0.55) = 0.75 + 0.1375 = 0.8875 \ \text{or 88.75%} \]