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Chapter 12: Problem 36

Foreign-language Study. Choose a student in a U.S. public high school at random and ask if he or she is studying a language other than English. Here is the distribution of results: \begin{tabular}{|l|c|c|c|c|c|} \hline Language & Sparish & French & German & All athers & None \\ \hline Probability & \(0.30\) & \(0.08\) & \(0.02\) & \(0.03\) & \(0.57\) \\ \hline \end{tabular} a. Explain why this is a legitimate probability model. b. What is the probability that a randomly chosen student is studying a language other than English? c. What is the probability that a randomly chosen student is studying French, German, or Spanish?

Short Answer

Expert verified
a. Legitimate model; b. Probability: 0.43; c. Probability: 0.40.

Step by step solution

01

Verify Total Probability

To check if this is a legitimate probability model, verify that the sum of all probabilities equals 1. Add all the given probabilities: \(0.30 + 0.08 + 0.02 + 0.03 + 0.57 = 1.00\). Since the total is 1, it confirms the model is legitimate.
02

Calculate Probability of Studying a Language Other Than English

To find the probability that a student is studying a language other than English, add the probabilities of all languages listed: \(0.30\) (Spanish), \(0.08\) (French), \(0.02\) (German), and \(0.03\) (All Others). Therefore, the probability is: \(0.30 + 0.08 + 0.02 + 0.03 = 0.43\).
03

Calculate Probability of Studying Specific Languages

To determine the probability that a student is studying French, German, or Spanish, add the probabilities for these specific languages: \(0.30\) (Spanish), \(0.08\) (French), and \(0.02\) (German). The total probability is: \(0.30 + 0.08 + 0.02 = 0.40\).

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Key Concepts

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

Foreign Language Study
Studying a foreign language can be very enriching. It not only helps in understanding different cultures but also enhances cognitive abilities. When we talk about foreign language study in the context of U.S. public high schools, the probability distribution indicates how many students are learning languages other than English. Different languages like Spanish, French, and German have different probabilities associated with them. This reflects how common or uncommon it is for students to study these languages. For example, a probability of 0.30 for Spanish means that if you randomly picked students, about 30% would be studying Spanish. This statistical insight helps schools understand the language preferences among students.
Legitimate Probability
A probability model is legitimate if all the probabilities add up to one. This is very important in any statistical analysis. To determine legitimacy, you need to tally all the probabilities assigned to each event or outcome. In our high school language study exercise:
  • Spanish: 0.30
  • French: 0.08
  • German: 0.02
  • All others: 0.03
  • None: 0.57
Adding these numbers (0.30 + 0.08 + 0.02 + 0.03 + 0.57 = 1.00), we see that it sums up to 1. This confirms that our probability model is legitimate. Each probability represents a fraction of the whole, and together they cover all possible outcomes.
Random Selection
When we pick a student at random, it means every student has an equal chance of being chosen. This is an essential part of probability to ensure that the sample is unbiased. Random selection is a common practice to understand trends, like the preference for studying languages other than English. In the exercise, selecting a student randomly is a fair method to estimate the probability of studying Spanish, French, German, or another language.
High School Statistics
Statistics are incredibly useful in understanding complex data sets, such as those related to education. In high school statistics, students are taught to analyze and interpret data. They learn to find insights such as the probabilities of students learning different languages. Using statistics:
  • You can determine the most popular foreign language among students.
  • You can also analyze shifts in language learning across years.
  • It provides a broader perspective on educational preferences and challenges.
These skills are not only crucial for academic success but also for making informed decisions later in life.

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Most popular questions from this chapter

What Type of Probability? (optional topic) The NASA website on global climate change says that "The current warming trend is of particular significance because most of it is extremely likely (greater than \(95 \%\) probability) to be the result of human activity since the mid-20th century."20 This probability is based on satellite data collecting many different types of information, historical data, scientific theory, and sophisticated computer models implementing the latest theory. What type of probability is this? Is it a probability based on the proportion of times an outcome would occur in a very long series of repetitions or a personal probability?

Will You Be in a Crash? The probability that a randomly chosen driver will be involved in a car crash in the next year is about \(0.051 . \underline{13}\) This is based on the proportion of millions of drivers who have crashes. a. What do you think is your own probability of being in a crash in the next year? This is a personal probability. b. Give some reasons why your personal probability might be a more accurate prediction of your "true chance" of being in a crash than the probability for a random driver. c. Almost everyone says their personal probability is lower than the random driver probability. Why do you think this is true?

How Many Cups of Coffee? Choose an adult age 18 or over in the United States at random and ask, "How many cups of coffee do you drink on average per day?" Call the response \(X\) for short. Based on a large sample survey, here is a probability model for the answer you will get: 8 \begin{tabular}{|c|c|c|c|c|c|} \hline Number & 0 & 1 & 2 & 3 & 4 or more \\ \hline Probability & \(0.36\) & \(0.26\) & \(0.19\) & \(0.08\) & \(0.11\) \\ \hline \end{tabular} a. Verify that this is a valid finite probability model. b. Describe the event \(X<4\) in words. What is \(P(X<4)\) ? c. Express the event "have at least one cup of coffee on an average day" in terms of \(X\). What is the probability of this event?

Door Prize. A party host gives a door prize to one guest chosen at random. There are 48 men and 42 women at the party. What is the probability that the prize goes to a woman? Explain how you arrived at your answer.

Languages in Canada. Canada has two official languages: English and French. Choose a Canadian at random and ask, "What is your mother tongue?" Here is the distribution of responses, combining many separate languages: \(\underline{5}\) \begin{tabular}{|l|c|c|c|} \hline Language & English & French & Other \\ \hline Probability & \(0.57\) & \(0.21\) & \(?\) \\ \hline \end{tabular} a. What is the probability that a Canadian's mother tongue is either English or French? b. What probability should replace "?" in the distribution? c. What is the probability that a Canadian's mother tongue is not English?
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