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Chapter 3: Problem 14

Use the following information to answer the next six exercises. There are 23 countries in North America, 12 countries in South America, 47 countries in Europe, 44 countries in Asia, 54 countries in Africa, and 14 in Oceania (Pacific Ocean region). Let A = the event that a country is in Asia. Let E = the event that a country is in Europe. Let F = the event that a country is in Africa. Let N = the event that a country is in North America. Let O = the event that a country is in Oceania. Let S = the event that a country is in South America. Find P(F).

Short Answer

Expert verified
P(F) = 54/194.

Step by step solution

01

Identify the Total Number of Countries

First, we need to find the total number of countries across all regions provided. Add the number of countries from each continent or region: North America has 23, South America has 12, Europe has 47, Asia has 44, Africa has 54, and Oceania has 14.
02

Calculate the Total Number of Countries

Add the numbers from each region to find the total count: \[ 23 + 12 + 47 + 44 + 54 + 14 = 194 \]So, there are 194 countries in total.
03

Identify the Number of Countries in Africa

According to the information given, there are 54 countries in Africa. The event F, the event that a country is in Africa, corresponds to this number.
04

Calculate the Probability of Event F

The probability of selecting a country from Africa, event F, is calculated as the ratio of the number of countries in Africa to the total number of countries. Thus, \[ P(F) = \frac{54}{194} \]

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Key Concepts

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

Understanding Event Probability
Event probability is a fundamental concept in probability theory. It helps us calculate the likelihood of a certain event happening. In our context, an event is a specific outcome, such as picking a country from a particular continent.
To find the probability of an event, we need two main pieces of information: the number of favorable outcomes and the total number of possible outcomes. The probability is then calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
  • The favorable outcome is the event we are interested in, such as selecting a country from Africa.
  • The total possible outcomes are all the countries we consider, in this case, every country in the provided regions.
Considering the probability of event F (selecting a country in Africa), we identified 54 countries as favorable outcomes, and 194 total countries as possible outcomes.
Using the Counting Principle
The counting principle is a helpful tool that simplifies determining the total number of outcomes. It allows us to systematically count all possibilities without missing any.
When applying the counting principle, we simply add up all individual counts of countries in each region to find the total number of countries. This straightforward addition is critical because accurately knowing the total number of possibilities ensures we can correctly calculate probabilities.
  • North America: 23 countries
  • South America: 12 countries
  • Europe: 47 countries
  • Asia: 44 countries
  • Africa: 54 countries
  • Oceania: 14 countries
The total number of countries, calculated using the counting principle, is 194.
Importance of Regional Distribution
Regional distribution helps us understand the geographical spread of outcomes. In probability problems, this is vital because it determines the set of possibilities we are working with.
Let's look at our geographical distribution: each region has a different number of countries. Understanding this distribution allows us to focus on specific regions when calculating probabilities since each one represents a different set of conditional probabilities.
  • The distribution shows that Africa has the highest number of countries (54), influencing its probability of selection.
  • This distribution helps identify specific regions more likely or less likely based on their count.
Recognizing these distributions can often reveal insights about factors affecting probabilities, such as size or location.
Executing Probability Calculation
Probability calculation is the process of determining how likely an event is to occur. We treat probability as a ratio, connected to the conceptual framework of favorable to total outcomes.
To compute the probability, we use the formula: \[P( ext{specific event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\]
For event F, selecting a country in Africa, the probability becomes:
  • Favorable outcomes: 54 (countries in Africa)
  • Total possible outcomes: 194 (all countries)
So, the probability is:\[P(F) = \frac{54}{194}\approx 0.278\]The result tells us that there's about a 27.8% chance to randomly select a country from Africa, highlighting the practical application of probability calculations.

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Most popular questions from this chapter

Use the following information to answer the next seven exercises. An article in the New England Journal of Medicine, reported about a study of smokers in California and Hawaii. In one part of the report, the self-reported ethnicity and smoking levels per day were given. Of the people smoking at most ten cigarettes per day, there were 9,886 African Americans, 2,745 Native Hawaiians, 12,831 Latinos, 8,378 Japanese Americans, and 7,650 Whites. Of the people smoking 11 to 20 cigarettes per day, there were 6,514 African Americans, 3,062 Native Hawaiians, 4,932 Latinos, 10,680 Japanese Americans, and 9,877 Whites. Of the people smoking 21 to 30 cigarettes per day, there were 1,671 African Americans, 1,419 Native Hawaiians, 1,406 Latinos, 4,715 Japanese Americans, and 6,062 Whites. Of the people smoking at least 31 cigarettes per day, there were 759 African Americans, 788 Native Hawaiians, 800 Latinos, 2,305 Japanese Americans, and 3,970 Whites. Find the probability that the person was Latino.

Q and R are independent events. P(Q) = 0.4 and P(Q AND R) = 0.1. Find P(R)

Consider the following scenario: Let P(C) = 0.4. Let P(D) = 0.5. Let P(C|D) = 0.6. a. Find P(C AND D). b. Are C and D mutually exclusive? Why or why not? c. Are C and D independent events? Why or why not? d. Find P(C OR D). e. Find P(D|C)

United Blood Services is a blood bank that serves more than 500 hospitals in 18 states. According to their website, a person with type O blood and a negative Rh factor (Rh-) can donate blood to any person with any bloodtype. Their data show that 43% of people have type O blood and 15% of people have Rh- factor; 52% of people have type O or Rh- factor. a. Find the probability that a person has both type O blood and the Rh- factor. b. Find the probability that a person does NOT have both type O blood and the Rh- factor.

Use the following information to answer the next seven exercises. An article in the New England Journal of Medicine, reported about a study of smokers in California and Hawaii. In one part of the report, the self-reported ethnicity and smoking levels per day were given. Of the people smoking at most ten cigarettes per day, there were 9,886 African Americans, 2,745 Native Hawaiians, 12,831 Latinos, 8,378 Japanese Americans, and 7,650 Whites. Of the people smoking 11 to 20 cigarettes per day, there were 6,514 African Americans, 3,062 Native Hawaiians, 4,932 Latinos, 10,680 Japanese Americans, and 9,877 Whites. Of the people smoking 21 to 30 cigarettes per day, there were 1,671 African Americans, 1,419 Native Hawaiians, 1,406 Latinos, 4,715 Japanese Americans, and 6,062 Whites. Of the people smoking at least 31 cigarettes per day, there were 759 African Americans, 788 Native Hawaiians, 800 Latinos, 2,305 Japanese Americans, and 3,970 Whites. Complete the table using the data provided. Suppose that one person from the study is randomly selected. Find the probability that person smoked 11 to 20 cigarettes per day.
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