Problem 39 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In a recent senatorial election, \(50 \%\) of the voters in a certain district were registered as Democrats, \(35 \%\) were registered as Republicans, and \(15 \%\) were registered as Independents. The incumbent Democratic senator was reelected over her Republican and Independent opponents. Exit polls indicated that she gained \(75 \%\) of the Democratic vote, \(25 \%\) of the Republican vote, and \(30 \%\) of the Independent vote. Assuming that the exit poll is accurate, what is the probability that a vote for the incumbent was cast by a registered Republican? -

Step-by-Step Solution

Answer

The probability that a vote for the incumbent was cast by a registered Republican is approximately \(17.24\% \).

Step 1: Identify probabilities

First, we need to identify the probabilities given in the problem. We have: - P(Democrat) = 0.50 - P(Republican) = 0.35 - P(Independent) = 0.15 - P(Incumbent | Democrat) = 0.75 - P(Incumbent | Republican) = 0.25 - P(Incumbent | Independent) = 0.30

Step 2: Use Bayes' theorem to find the probability

According to Bayes' theorem, we have: P(Republican | Incumbent) = P(Incumbent | Republican) * P(Republican) / P(Incumbent) We have P(Incumbent | Republican) and P(Republican), but we still need to find P(Incumbent). To find P(Incumbent), we can use the law of total probability: P(Incumbent) = P(Incumbent | Democrat) * P(Democrat) + P(Incumbent | Republican) * P(Republican) + P(Incumbent | Independent) * P(Independent) Plugging in the values, we get: P(Incumbent) = (0.75 * 0.50) + (0.25 * 0.35) + (0.30 * 0.15) = 0.375 + 0.0875 + 0.045 = 0.5075 Now we can use Bayes' theorem to find P(Republican | Incumbent): P(Republican | Incumbent) = (0.25 * 0.35) / 0.5075 = 0.0875 / 0.5075 = 0.172414

Step 3: Convert the probability into a percentage

Finally, we will convert the probability into a percentage: Probability = 0.172414 * 100% = 17.2414 % Therefore, the probability that a vote for the incumbent was cast by a registered Republican is approximately 17.24%.

Key Concepts

Probability

Probability is a fundamental concept used to quantify uncertainty. It measures the likelihood of different outcomes occurring. In simple terms, probability can be seen as the chance of something happening. It's calculated as a ratio between favorable outcomes and the total number of possible outcomes.

In the context of the election problem, probabilities are assigned to the registration status of voters. For example:

• P(Democrat) = 0.50 means there's a 50% chance that a randomly chosen voter is a Democrat.
• P(Republican) = 0.35 indicates a 35% probability for a voter being Republican.
• P(Independent) = 0.15 shows a 15% likelihood for a voter being Independent.

By properly understanding and calculating probabilities, we get a clearer picture of the context and outcomes, such as predicting election results.

Conditional Probability

Conditional probability builds on basic probability by focusing on the likelihood of an event given the occurrence of another event. It’s symbolically written as \( P(A | B) \), which is read as "the probability of A given B."

In the election example, we consider conditional probabilities of the incumbent senator winning the vote based on the voter’s party affiliations. For instance:

• P(Incumbent | Democrat) = 0.75: 75% of Democrats voted for the incumbent.
• P(Incumbent | Republican) = 0.25: Only 25% of Republicans voted for her.
• P(Incumbent | Independent) = 0.30: 30% of Independents supported her.

Conditional probability helps specify probabilities in more complex scenarios by considering the effects of additional information, making it a powerful tool in analyses such as election polling.

Election Analysis

Election analysis often uses Bayes' Theorem to derive useful insights from poll data. Bayes’ Theorem allows us to update probabilities based on new evidence, and it is written as: \( P(A | B) = \frac{P(B | A) \cdot P(A)}{P(B)} \).

In our election case, we aim to determine the probability that a Republican voter actually voted for the incumbent. By plugging in the known values:

• Probability of casting a vote for the incumbent given the voter is Republican: P(Incumbent | Republican) = 0.25
• Probability a voter is Republican: P(Republican) = 0.35
• Total probability of voting for the incumbent (from all affiliations combined) is computed using the law of total probability: P(Incumbent) = 0.5075

Finally, using Bayes' Theorem, the probability that a vote for the incumbent came from a Republican is calculated as \( P(Republican | Incumbent) = \frac{0.25 \times 0.35}{0.5075} \) which equals roughly 17.24%.

Thus, Bayes' Theorem and probabilities together provide a vital framework to interpret election data more clearly and make informed predictions.