91Ó°ÊÓ

(Optional topic) Should the government help the poor? In the 2014 General Social Survey, \(32 \%\) of those sampled thought of themselves as Democrats, \(45 \%\) as Independents, \(21 \%\) as Republicans, and \(2 \%\) as Other. \({ }^{23}\) When asked, " Should the government in Washington do everything possible to improve the standard of living of all poor Americans?"" \(23 \%\) of the Democrats, \(18 \%\) of the Independents, \(4 \%\) of the Republicans, and \(15 \%\) of Others agreed. Given that a person agrees that the government in Washington should do everything possible to improve the standard of living of all poor Americans, use Bayes' rule to find the probability the person thinks of him- or herself as a Democrat.

Step by step solution

01

Identify the events

Let \( D \) be the event that a person thinks of themselves as a Democrat. Let \( A \) be the event that a person agrees the government should help the poor.

02

Determine Given Probabilities

Given probabilities from the problem are as follows:\ \( P(D) = 0.32 \) (Probability of being a Democrat),\ \( P(A|D) = 0.23 \) (Probability of agreeing if Democrat).

03

Determine Probability of Agreement

Calculate \( P(A) \), the probability that a person agrees the government should help the poor.\ \[ P(A) = P(A|D)\cdot P(D) + P(A|I)\cdot P(I) + P(A|R)\cdot P(R) + P(A|O)\cdot P(O) \]\ where \( I,R,O \) are Independents, Republicans, and Others respectively: \( P(I) = 0.45,\ P(R) = 0.21,\ P(O) = 0.02 \), \( P(A|I) = 0.18,\ P(A|R) = 0.04,\ P(A|O) = 0.15 \).

04

Compute Probability of Agreement

Substitute the values calculated in Step 3 into the expression for \( P(A) \):\ \[ P(A) = 0.23\cdot 0.32 + 0.18\cdot 0.45 + 0.04\cdot 0.21 + 0.15\cdot 0.02 \].\ \[ P(A) = 0.0736 + 0.081 + 0.0084 + 0.003 = 0.166 \].

05

Apply Bayes' Rule

Find \( P(D|A) \) using Bayes' Rule:\ \[ P(D|A) = \frac{P(A|D)\cdot P(D)}{P(A)} \].\ Substituting the known values gives\ \[ P(D|A) = \frac{0.23\cdot 0.32}{0.166} = \frac{0.0736}{0.166} \approx 0.443 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Probability Calculation

Probability calculation is the foundational step in determining the likelihood of an event occurring. The essence of probability is to quantify the chance of an event, expressed as a number between 0 and 1. Let’s break down how probabilities are calculated in a scenario:

• Firstly, define the events clearly. In our exercise, events include being a Democrat, Independent, Republican, or Another party, and agreeing with a governmental policy.
• Next, identify known probabilities. For instance, the straight probabilities: 32% of those asked are Democrats.
• Combine probabilities when necessary. In this task, we compute the overall probability of agreement that the government should help the poor by considering the probabilities of agreement within each political group.

It’s like crafting a recipe, where each ingredient’s proportion contributes to the overall flavor. Always ensure your probabilities add up appropriately, helping to reliably predict the likelihood of subsequent events.

Exploring Conditional Probability

Conditional probability extends the concept of classic probability by focusing on the likelihood of an event occurring given that another event has already happened. Essentially, it's a way to refine probability calculations when you have additional information.
For instance, in the exercise, we looked at the probability of someone agreeing that the government should help the poor if we already know they're a Democrat. This is represented mathematically as \( P(A|D) \), meaning the probability of agreement \( A \) given Democrat \( D \).

• Conditional probabilities are central in contexts where multiple events might influence an outcome.
• It uses known probabilities in a narrower scope to derive new insight.
• It requires careful combination of original probabilities to yield new, context-specific data.

Think of it as seeing through a clear lens, where knowing one detail helps us see the bigger picture more accurately.

Applying Statistical Reasoning

Statistical reasoning allows us to make sense of numbers and probabilities, transforming them into actionable insights. It involves using probability data to derive logical conclusions about the world.
In our exercise, statistical reasoning is applied using Bayes' Theorem. This theorem helps update our belief about a hypothesis, in this case, how likely someone is to be a Democrat if they support governmental aid for the poor, based on prior knowledge.

• Bayes' Theorem combines our prior knowledge with new evidence. For example, using known data about political affiliations and agreement rates.
• It’s an iterative process, constantly refining understanding by incorporating new information.
• In practice, statistical reasoning solidifies abstract probability into real-world applications, making seemingly vague data meaningful.

Consider it a wise teacher, guiding you from assumptions and raw data to informed conclusions, showing how interconnected information leads to new insights.