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Chapter 5: Problem 67

Foreign-language study Choose a student in grades 9 to 12 at random and ask if he or she is studying a language other than English. Here is the distribution of results: $$\begin{array}{lccccc}\hline \text { Language: } & \text { Spanish } & \text { French } & \text { German } & \text { All others } & \text { None } \\\\\text { Probability: } & 0.26 & 0.09 & 0.03 & 0.03 & 0.59 \\\\\hline\end{array}$$ (a) What's the probability that the student is studying a language other than English? (b) What is the conditional probability that a student is studying Spanish given that he or she is studying some language other than English?

Short Answer

Expert verified
a) 0.41 b) 0.63

Step by step solution

01

Understand the Probabilities

We are given the probabilities of students studying different languages and none at all. The probabilities are: Spanish = 0.26, French = 0.09, German = 0.03, All others = 0.03, and None = 0.59.
02

Find Probability of Studying Any Language

To find the overall probability of a student studying a language other than English, sum the probabilities of Spanish, French, German, and all other languages: \(0.26 + 0.09 + 0.03 + 0.03\).
03

Calculate the Total Other-than-English Probability

Compute the sum: \(0.26 + 0.09 + 0.03 + 0.03 = 0.41\). Thus, the probability of studying any language other than English is \(0.41\).
04

Conditional Probability Formula

The conditional probability of an event A given an event B is given by \(P(A|B) = \frac{P(A \cap B)}{P(B)}\). Here, A is the event of studying Spanish, and B is the event of studying a language other than English.
05

Calculate Conditional Probability of Spanish

The probability of studying Spanish (A) is 0.26, and the probability of studying a language other than English (B) is 0.41. The intersection of A and B is simply the probability of A, as A is a subset of B. Thus, \(P(\text{Spanish} | \text{Other than English}) = \frac{0.26}{0.41}\).
06

Compute the Conditional Probability

Divide 0.26 by 0.41 to get the conditional probability: \(\frac{0.26}{0.41} \approx 0.634\).

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Key Concepts

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

Conditional Probability
Conditional probability is a key concept when dealing with events that rely on each other. It helps us determine the probability of an event occurring, given that another event has already happened. For example, consider a case where we want to find the chance of a student studying Spanish, knowing they already study a language other than English.
To calculate conditional probability, we use the formula: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \] Where:
  • \(P(A|B)\) is the probability of event A occurring given event B occurs.
  • \(P(A \cap B)\) is the probability that both events A and B occur.
  • \(P(B)\) is the probability of event B.
In our example, event A is studying Spanish and event B is studying any language other than English. This means we use the given probability of studying Spanish over the probability of studying any non-English language.
Random Selection
Random selection is a method used to ensure that every individual has an equal chance of being chosen. This principle strengthens the reliability of statistical results. In the context of our example, we randomly select a student from grades 9 to 12 to inquire about their language study.
When selection is random, we trust that the sample is unbiased and truly represents the population. This trustworthiness is crucial because it affects the validity of the probabilities and conclusions drawn from the data.
For instance, if we randomly choose from all students, any specific student's chance of inclusion in our language study is unaffected by external influences. Hence, conclusions drawn regarding language study statistics reflect the genuine tendencies of the student body.
Probability Distribution
A probability distribution provides a model that indicates the likelihood of different outcomes in a random experiment. It lists all possible events and their respective probabilities, giving a complete picture of how probabilities are distributed.
In the language study example, the distribution is expressed through:
  • Spanish: 0.26
  • French: 0.09
  • German: 0.03
  • All other languages: 0.03
  • None: 0.59
The sum of these probabilities equals 1, signifying a comprehensive account of all potential outcomes. Understanding these probabilities helps us to deduce the likelihood that a student selected at random speaks a specific language or no language at all.
Language Study Statistics
Language study statistics break down how many students are engaged in studying different languages. It sheds light on the educational trends and preferences of students.
The statistics reveal that studying Spanish is the most common choice among students in grades 9 to 12, followed by French and German. This observation aligns with broader educational trends witnessed in many schools where Spanish is prioritized due to practical usage and demand.
By analyzing statistics, educators can determine which languages might need more resources or encouragement. It also helps them to understand shifts in language importance across different grades and adjust curricula accordingly.

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Most popular questions from this chapter

Lost Internet sites Internet sites often vanish or move, so that references to them can't be followed. In fact, \(13 \%\) of Internet sites referenced in major scientific journals are lost within two years after publication. \({ }^{22}\) If we randomly select seven Internet references, from scientific journals, what is the probability that at least one of them doesn't work two years later?

Fill 'er up! In a recent month, \(88 \%\) of automobile drivers filled their vehicles with regular gasoline, \(2 \%\) purchased midgrade gas, and \(10 \%\) bought pre- mium gas. \({ }^{18}\) Of those who bought regular gas, \(28 \%\) paid with a credit card; of customers who bought midgrade and premium gas, \(34 \%\) and \(42 \%\), respectively, paid with a credit card. Suppose we select a customer at random. (a) Draw a tree diagram to represent this situation. (b) Find the probability that the customer paid with a credit card. Show your work. (c) Given that the customer paid with a credit card, find the probability that she bought premium gas. Show your work.

A basketball player claims to make \(47 \%\) of her shots from the field. We want to simulate the player taking sets of 10 shots, assuming that her claim is true. To simulate the number of makes in 10 shot attempts, you would perform the simulation as follows: (a) Use 10 random one-digit numbers, where \(0-4\) are a make and \(5-9\) are a miss. (b) Use 10 random two-digit numbers, where \(00-46\) are a make and \(47-99\) are a miss. (c) Use 10 random two-digit numbers, where \(00-47\) are a make and \(48-99\) are a miss. (d) Use 47 random one-digit numbers, where 0 is a make and \(1-9\) are a miss. (e) Use 47 random two-digit numbers, where \(00-46\) are a make and \(47-99\) are a miss.

Taking the train According to New Jersey Transit, the 8: 00 A.M. weekday train from Princeton to New York City has a \(90 \%\) chance of arriving on time. To test this claim, an auditor chooses 6 weekdays at random during a month to ride this train. The train arrives late on 2 of those days. Does the auditor have convincing evidence that the company's claim isn't true? Design and carry out a simulation to estimate the probability that a train with a \(90 \%\) chance of arriving on time each day would be late on 2 or more of 6 days. Follow the four-step process.

Brushing teeth, wasting water? A recent study reported that fewer than half of young adults turn off the water while brushing their teeth. Is the same true for teenagers? To find out, a group of statistics students asked an SRS of 60 students at their school if they usually brush with the water off. In the sample, 27 students said "No." The Fathom dotplot below shows the results of taking 200 SRSs of 60 students from a population in which the true proportion who brush with the water off is 0.50 . (a) Explain why the sample result does not give convincing evidence that fewer than half of the school's students brush their teeth with the water off. (b) Suppose instead that 18 students in the class's sample had said "No." Explain why this result would give strong evidence that fewer than \(50 \%\) of the school's students brush their teeth with the water off.
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