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Chapter 6: Problem 65

According to the Sydney Morning Herald, \(40 \%\) of bicycles stolen in Holland are recovered. (In contrast, only \(2 \%\) of bikes stolen in New York City are recovered.) Find the probability that, in a sample of 6 randomly selected cases of bicycles stolen in Holland, exactly 2 out of 6 bikes are recovered.

Short Answer

Expert verified
The probability that, out of 6 randomly selected bicycle theft cases in Holland, exactly 2 bikes are recovered is approximately 0.276 or 27.6%.

Step by step solution

01

Identify the numbers to use in the formula

The number of trials \(n\) is 6 (bikes stolen in Holland), the number of successes \(x\) is 2 (bikes recovered) and the probability of success \(p\) is 0.4 (40% recovery rate)
02

Calculation of combinations

The term \(C(n, x)\) in the binomial probability formula represents combinations. It can be calculated as \(C(n, x) = \frac{n!}{x!(n-x)!}\), where \(!\) stands for factorial. Replacing \(n=6\) and \(x=2\) into this formula, we get \(C(6, 2) = \frac{6!}{2!(6-2)!}= 15\)
03

Applying the binomial probability formula

Now replace the calculated and given values into the binomial formula: \(P(x) = C(n, x) \cdot (p^x) \cdot ((1-p)^{n-x}) = 15 \cdot (0.4^2) \cdot ((1-0.4)^{6-2}) =15 \cdot 0.16 \cdot (0.6)^4 = 0.27648\)
04

Finalize the result

The probability that, in a sample of 6 randomly selected bicycle theft cases in Holland, exactly 2 out of 6 bikes are recovered is 0.27648

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Key Concepts

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

Probability Theory
Probability theory is a branch of mathematics that deals with the analysis of random events. The outcome of a random event cannot be determined before it occurs, but it may be any one of several possible outcomes. The actual outcome is considered to be determined by chance.

The notion of probability can be quantified through various models, including the frequency of occurrences or the subjective belief about the likelihood of an occurrence. In the case of the Sydney Morning Herald's report on bicycle theft recovery rates in Holland, probability theory is utilized to compute the likelihood of a certain number of recovered bicycles out of a sample.
Combinations
In mathematics, combinations refer to the selection of items from a collection, such that the order of selection does not matter. This concept is often used in probability theory to calculate the number of ways events can occur.

For example, if you want to know how many different ways there are to select 2 bicycles out of 6 to be recovered, you would calculate the combinations of 6 taken 2 at a time, symbolized by \( C(6, 2) \). This calculation is essential for solving problems involving probability where specific outcomes from a group are required.
Factorial
The factorial function (denoted with an exclamation point '!' after a number) is a function that multiplies a series of descending natural numbers. Essentially, \( n! \) means the product of all positive integers less than or equal to \( n \).

For example, \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \). Factorials are particularly useful in probability theory when calculating combinations, as they are part of the formula to determine the number of possible ways in which events can occur without regard to the order.
Statistics Education
Statistics education is crucial for equipping students with the ability to analyze, interpret, and present data effectively. A sound understanding of statistics is essential not only in the field of mathematics but also in various domains such as science, social science, and business.

In statistics education, concepts like binomial probability must be taught with practical examples and exercises, such as computing the probability of recovering a certain number of stolen bikes. This hands-on approach helps students apply mathematical theory to real-life problems and enhances their analytical skills.

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Most popular questions from this chapter

Suppose the probability that a randomly selected couple opens a joint bank account within 1 year of marriage is \(0.3\), and the probability that a randomly selected couple opens a joint bank account after a year of marriage is 0.4. Take a random sample of 15 people opening their accounts within 1 year and 15 people opening their accounts after 1 year. The sample chosen is such that either the husband or the wife is included in it. Why is the binomial model inappropriate for finding the probability that exactly 8 out of the 30 people in the sample will open joint bank accounts within 1 year? List all of the binomial conditions that are not met.

Stanford-Binet IQs for children are approximately Normally distributed and have \(\mu=100\) and \(\sigma=15 .\) What is the probability that a randomly selected child will have an IQ of 115 or above?

A study of U.S. births published on the website Medscape from WebMD reported that the average birth length of babies was \(20.5\) inches and the standard deviation was about \(0.90\) inch. Assume the distribution is approximately Normal. Find the percentage of babies who have lengths of 19 inches or less at birth.

You may have heard that drunk driving is dangerous, but what about drunk walking? According to federal information (reported in the Ventura County Star on August \(6 .\) 2013 ), \(50 \%\) of the pedestrians killed in the United States had a blood-alcohol level of \(0.08 \%\) or higher. Assume that two randomly selected pedestrians who were killed are studied. a. If the pedestrian was drunk (had a blood-alcohol level of \(0.08 \%\) or higher), we will record a D, and if the pedestrian was not drunk, we will record an \(\mathrm{N}\). List all possible sequences of \(\mathrm{D}\) and \(\mathrm{N}\). b. For each sequence, find the probability that it will occur, by assuming independence. c. What is the probability that neither of the two pedestrians was drunk? d. What is the probability that exactly one out of two independent pedestrians was drunk? c. What is the probability that both were drunk?

According to National Vital Statistics, the average length of a newborn baby is \(19.5\) inches with a standard deviation of \(0.9\) inch. The distribution of lengths is approximately Normal. Use a table or technology for each question. Include an appropriately labeled and shaded Normal curve for each part. There should be three separate curves. an What is the probability that a baby will have a length of \(20.4\) inches or more? b. What is the probability that a baby will have a length of \(21.4\) inches or more? c. What is the probability that a baby will be between 18 and 21 inches in length?
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