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Who Takes the GMAT? In many settings, the "rules of probability" are just basic facts about percents. The Graduate Management Admission Test (GMAT) website provides the following information about the geographic region of citizenship of those who took the test in \(2015: 2.1 \%\) were from Africa; \(0.4 \%\) were from Australia and the Pacific Islands; \(2.5 \%\) were from Canada; \(12.9 \%\) were from Central and South Asia; \(32.0 \%\) were from East and Southeast Asja; 2.0\% were from Eastern Europe; \(3.2 \%\) were from Mexico, the Caribbean, and Latin America; \(3.0 \%\) were from the Middle East; \(34.2 \%\) were from the United States; and \(7.7 \%\) were from Western Europe.3 (a) What percent of those who took the test in 2015 were from North America (either Canada, the United States, Mexico, the Caribbean, or Latin America)? Which rule of probability did you use to find the answer? (b) What percent of those who took the test in 2015 were from some other region than the United States? Which rule of probability did you use to find the answer?

Step by step solution

01

Identifying North American Regions

North America comprises Canada, the United States, Mexico, the Caribbean, and Latin America. We need to add these percentages to find the answer.

02

Calculating Percent from North America

Add the percentages for Canada (2.5%), the United States (34.2%), and Mexico, the Caribbean, and Latin America (3.2%). \(2.5\% + 34.2\% + 3.2\% = 39.9\%\)

03

Identifying Regions Other Than the United States

To find the percentage from regions other than the United States, we need to add the percentages of all regions except the United States.

04

Calculating Percent from Other Regions

Find the sum of all other regions: \(2.1\% + 0.4\% + 2.5\% + 12.9\% + 32.0\% + 2.0\% + 3.2\% + 3.0\% + 7.7\%\)This equals \(65.8\%\).

05

Applying Probability Rule

For part (a), we used the additive rule of probability, where to find the probability of either of multiple events, we sum their individual probabilities. For part (b), we used the complementary rule, finding what was not in the United States (by subtracting the percentage for the United States from 100%).

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Key Concepts

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

Additive Rule of Probability

The Additive Rule of Probability is a simple yet powerful tool in probability theory. It is used to find the probability of any one of multiple events occurring. This rule is applicable to both mutually exclusive and non-mutually exclusive events.

When events are mutually exclusive (they cannot happen at the same time), the rule states that the probability of either event A or event B occurring is the sum of their individual probabilities:

• \( P(A \cup B) = P(A) + P(B) \)

For non-mutually exclusive events, you must adjust for any overlap between events. The formula becomes:

• \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \)

In our exercise, since the regions in North America cannot overlap (you cannot be from two different countries at once), we simply added their probabilities (percentages). This real-world application of the additive rule helps you logically break down large datasets into understandable segments.

Complementary Rule of Probability

The Complementary Rule of Probability is essential when you want to find out the probability of an event not happening. It's based on the concept that the sum of the probabilities of all possible outcomes of a particular event is one (or 100%).

The formula is:

• \( P( ext{{not A}}) = 1 - P(A) \)

This rule is particularly straightforward: if you know the probability of an event occurring, subtract it from one to get the probability of it not occurring.

In our GMAT exercise, we wanted to calculate the percentage of test takers from regions other than the United States. Here, the complementary rule was employed by subtracting the percentage of U.S. test-takers from 100% to get the percentage from other regions. Understanding this rule can save time by avoiding counting each possibility individually.

Geographic Data Analysis

Geographic Data Analysis in probability involves analyzing data that is grouped by geographic regions. This method provides insight into how different regions contribute to a dataset, like the population of GMAT test-takers by region.

Using geographic data helps identify patterns or trends. For instance, in our exercise, we could see a significant percentage of test-takers from East and Southeast Asia. Understanding these patterns is crucial for making informed decisions, whether in business, education, or public policy.

Analyzing geographic information often involves classifying data into discrete groups (such as continents or countries) and applying probability rules to derive meaningful insights. It's a way of contextualizing numbers by recognizing cultural, economic, and regional influences that might affect the data. Geographic data underscores the importance of both probability rules and data interpretation in presenting a holistic view of the analyzed dataset.