Problem 36 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In a past presidential election, it was estimated that the probability that the Republican candidate would be elected was \(\frac{3}{5}\), and therefore the probability that the Democratic candidate would be elected was \(\frac{2}{5}\) (the two Independent candidates were given no chance of being elected). It was also estimated that if the Republican candidate were elected, the probability that a conservative, moderate, or liberal judge would be appointed to the Supreme Court (one retirement was expected during the presidential term) was \(\frac{1}{2}, \frac{1}{3}\), and \(\frac{1}{6}\), respectively. If the Democratic candidate were elected, the probabilities that a conservative, moderate, or liberal judge would be appointed to the Supreme Court would be \(\frac{1}{8}, \frac{3}{8}\), and \(\frac{1}{2}\), respectively. A conservative judge was appointed to the Supreme Court during the presidential term. What is the probability that the Democratic candidate was elected?

Step-by-Step Solution

Answer

The probability that the Democratic candidate was elected given that a conservative judge was appointed to the Supreme Court is \(\frac{1}{14}\).

Step 1: Identify the Known Probabilities

There are several probabilities provided in the problem. The probability that the Republican candidate would be elected is \(\frac{3}{5}\), and the probability that the Democratic candidate would be elected is \(\frac{2}{5}\). Lastly, the probabilities of appointing different types of judges are given for both candidates. We are specifically interested in the probabilities associated with appointing a conservative judge.

Step 2: Identify the Events and their Probabilities

Define the following events. Let's call D the event that the Democratic candidate is elected (probability \(P(D)=\frac{2}{5}\)), and C be the event that a conservative judge is appointed. The probability that a conservative judge is appointed given that the Democratic candidate is elected is \(P(C|D)=\frac{1}{8}\).

Step 3: Calculate the Total Probability of Appointing a Conservative Judge

The total probability of appointing a conservative judge would be the sum of the probabilities of appointing a conservative judge given each candidate is elected, weighted by the probability of each candidate being elected. This would be: \[P(C) = P(D) \times P(C|D) + P(R) \times P(C|R)\] where R is the event that the Republican candidate is elected (\(P(R)=\frac{3}{5}\)), and \(P(C|R)=\frac{1}{2}\). Substituting the given values, we get: \[P(C) = \frac{2}{5} \times \frac{1}{8} + \frac{3}{5} \times \frac{1}{2} = \frac{7}{20}\]

Step 4: Use Bayes' Theorem to Find the Desired Probability

Finally, use Bayes' Theorem to find the probability that the Democratic candidate is elected given that a conservative judge is appointed. Bayes' Theorem is: \[P(D|C) = \frac{P(D) \times P(C|D)}{P(C)}\] Substituting the values we have found, we get: \[P(D|C) = \frac{\frac{2}{5} \times \frac{1}{8}}{\frac{7}{20}} = \frac{1}{14}\] Therefore, the probability that the Democratic candidate was elected given that a conservative judge was appointed to the Supreme Court is \(\frac{1}{14}\).

Key Concepts

Bayes' Theorem

Bayes' Theorem is a fundamental concept in probability theory that allows us to update the probability of a hypothesis as more evidence or information becomes available. In our exercise, Bayes' Theorem helps us calculate the probability that the Democratic candidate was elected, given that a conservative judge was appointed. This involves understanding conditional probability, which is the likelihood of an event occurring given that another event has already occurred. The theorem is expressed in the formula:\[ P(A|B) = \frac{P(A) \cdot P(B|A)}{P(B)} \]Where:- \(P(A|B)\) is the probability of event A given event B has occurred.- \(P(A)\) is the prior probability of event A.- \(P(B|A)\) is the likelihood of event B given A.- \(P(B)\) is the probability of event B occurring.In this case:

• \(P(D|C)\) represents the probability of the Democratic candidate being elected given a conservative judge was chosen.
• \(P(D)\) is the initial probability of the Democratic candidate being elected.
• \(P(C|D)\) is the probability of selecting a conservative judge if the Democratic candidate is elected.
• \(P(C)\) is the total probability of appointing a conservative judge.

By inputting these values, we discovered the probability that the Democratic candidate was elected, given the appointment of a conservative judge, was \( \frac{1}{14} \).

Probability Theory

Probability theory is the branch of mathematics concerned with analyzing random phenomena. It uses models and formulas to describe the likelihood of different possible outcomes. This exercise on predicting election outcomes and judicial appointments illustrates the use of probability theory. Probability of events such as a candidate winning or a judge being appointed are central to solving such problems.In this scenario, the probabilities are:

• The Republican candidate's election probability: \( \frac{3}{5} \)
• The Democratic candidate's election probability: \( \frac{2}{5} \)
• Appointment of a conservative judge if a Republican is elected: \( \frac{1}{2} \)
• Appointment of a conservative judge if a Democrat is elected: \( \frac{1}{8} \)

These probabilities form the foundation for further analysis using Bayes' Theorem, essentially linking probability theory to real-world contexts like elections. By combining these likelihoods, we derive the total chance of a conservative judge being appointed, which is crucial for calculating more complex conditional probabilities.

Election Prediction

Election prediction involves using statistical tools to forecast the outcome of an election based on various probabilities. In the exercise, predicting whether a Republican or Democratic candidate would win is treated as a probabilistic event. Initially, the Republican had a higher probability of winning (\( \frac{3}{5} \)) compared to the Democrat (\( \frac{2}{5} \)).Elections are an example of an event where predictions play a crucial role in preparing for political, economic, and social impacts. Bayesian analysis, as shown, extends these predictions by incorporating observable events, like the appointment of judges, to refine the probability of which candidate succeeded. This type of prediction is affected by:

• Historical voting patterns
• Current political climate
• Public opinion polls
• External factors such as economic conditions or international events

Understanding these factors helps analysts and statisticians provide more accurate predictions, adding another layer to the study of probability.

Mathematical Reasoning

Mathematical reasoning involves the logical thought process required to solve problems. It includes identifying known data, choosing appropriate methods, and methodically solving equations to find a solution. In our exercise, mathematical reasoning enabled us to:

• Identify relevant probabilities from the problem statement.
• Designate events like the election of candidates and judge appointments to solve the probability of interest.
• Apply Bayes' Theorem correctly to find the desired conditional probability.

This type of reasoning is critical in ensuring that each step in the solution process is validated and makes sense logically. By clearly defining each event and using correct mathematical principles, complex real-world problems become manageable. Furthermore, developing mathematical reasoning skills not only aids in problem-solving but also sharpens critical thinking, helping translate abstract mathematical concepts to practical applications.