Problem 18 - A First Course in Probability

Question

A total of 46 percent of the voters in a certain city classify themselves as Independents, whereas 30 percent classify themselves as Liberals and 24 percent say that they are Conservatives. In a recent local election, 35 percent of the Independents, 62 percent of the Liberals, and 58 percent of the Conservatives voted. A voter is chosen at random. Given that this person voted in the local election, what is the probability that he or she is (a) an Independent? (b) a Liberal? (c) a Conservative? (d) What fraction of voters participated in the local election?

Step-by-Step Solution

Answer

(a) The probability that a random person who voted is an Independent is approximately \(0.3309\). (b) The probability that a random person who voted is a Liberal is approximately \(0.3823\). (c) The probability that a random person who voted is a Conservative is approximately \(0.2868\). (d) The fraction of voters who participated in the local election is \(0.4862\) or approximately \(48.62\%\).

Step 1: Calculate joint probabilities of voting and being in each group

Now, we will calculate the joint probabilities of each group and voting, which can be obtained by multiplying the probabilities: P(I)P(V|I) = 0.46 * 0.35 = 0.161 P(L)P(V|L) = 0.30 * 0.62 = 0.186 P(C)P(V|C) = 0.24 * 0.58 = 0.1392

Step 2: Calculate total probability of voting

The total probability of voting, P(V), can be calculated by adding the joint probabilities obtained in step 1: P(V) = P(I)P(V|I) + P(L)P(V|L) + P(C)P(V|C) = 0.161 + 0.186 + 0.1392 = 0.4862

Step 3: Calculate conditional probabilities

Now, we will calculate the probabilities P(I|V), P(L|V), P(C|V) using the formula for conditional probability: P(I|V) = \(\frac{P(I)P(V|I)}{P(V)}\) = \(\frac{0.161}{0.4862}\) ≈ 0.3309 P(L|V) = \(\frac{P(L)P(V|L)}{P(V)}\) = \(\frac{0.186}{0.4862}\) ≈ 0.3823 P(C|V) = \(\frac{P(C)P(V|C)}{P(V)}\) = \(\frac{0.1392}{0.4862}\) ≈ 0.2868

Step 4: Present the results

Now, we have the probabilities for each question: (a) The probability that a random person who voted is an Independent is approximately 0.3309. (b) The probability that a random person who voted is a Liberal is approximately 0.3823. (c) The probability that a random person who voted is a Conservative is approximately 0.2868. (d) The fraction of voters who participated in the local election is 0.4862 or approximately 48.62%.

Key Concepts

Bayes' Theorem

Bayes’ Theorem is an important concept in probability theory that allows us to update our beliefs about an event based on new information. It's expressed in the formula:
\[ P(A|B) = \frac{P(B|A) \times P(A)}{P(B)} \]
Where:

• \(P(A|B)\) is the probability of event \(A\) occurring given that \(B\) is true.
• \(P(B|A)\) is the probability of event \(B\) given that \(A\) is true.
• \(P(A)\) and \(P(B)\) are the probabilities of events \(A\) and \(B\) independently of each other.

Bayes’ Theorem is incredibly useful for situations where you want to find a conditional probability. In the exercise, it's applied to determine the probability of a person being an Independent, Liberal, or Conservative given that they have voted. By plugging in the various known probabilities for each political group’s voting behavior and the overall voting probability, Bayes’ Theorem helps uncover the hidden likelihoods related to specific characteristics of the voters.

Joint Probability

Joint probability is a way to calculate the probability of two events happening at the same time. In mathematical terms, if you have two events \(A\) and \(B\), the joint probability is represented as \(P(A \cap B)\).
For instance, in the given exercise, we determine the joint probabilities of voters being from one of three political groups (Independents, Liberals, Conservatives) and also having voted. This is calculated by multiplying the probability of belonging to a group by the probability of voting within that group.

• \(P(I)P(V|I)\) signifies the joint probability of being an Independent and having voted.
• \(P(L)P(V|L)\) stands for the joint probability of being a Liberal and voting.
• \(P(C)P(V|C)\) is the joint probability of being a Conservative and participating in voting.

Understanding joint probabilities is crucial in scenarios where different factors or events might overlap or influence one another. Thus, it sets the groundwork for calculating total probabilities across an entire population.

Probability Theory

Probability Theory is a fundamental branch of mathematics focusing on the analysis of random phenomena. It's the backbone that supports concepts such as conditional probability, joint probability, and Bayes’ Theorem. Probability Theory provides the formal mathematical treatment for making quantifiable predictions based on statistical data.
In probability theory, two key types of probabilities are central:

• Marginal Probability: The probability of a single event occurring without consideration of any other events.
• Conditional Probability: The probability of an event occurring given that another event has already occurred, such as the likelihood of being a Liberal given that the person voted.

By using these concepts, probability theory allows us to make decisions or predictions about events based on incomplete or uncertain information. In the exercise, probability theory guides the process of calculating how many voters participated in the election, and the likelihood of their political affiliation based on that participation.