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A new analytical method to detect pollutants in water is being tested. This new method of chemical analysis is important because, if adopted, it could be used to detect three different contaminants-organic pollutants, volatile solvents, and chlorinated compounds-instead of having to use a single test for each pollutant. The makers of the test claim that it can detect high levels of organic pollutants with \(99.7 \%\) accuracy, volatile solvents with \(99.95 \%\) accuracy, and chlorinated compounds with \(89.7 \%\) accuracy. If a pollutant is not present, the test does not signal. Samples are prepared for the calibration of the test and \(60 \%\) of them are contaminated with organic pollutants, \(27 \%\) with volatile solvents, and \(13 \%\) with traces of chlorinated compounds. A test sample is selected randomly. a. What is the probability that the test will signal? b. If the test signals, what is the probability that chlorinated compounds are present?

Step by step solution

01

Calculate Probability of Signal Due to Organic Pollutants

The probability that a sample is contaminated with organic pollutants is given as 0.6, and the probability of detecting it when present is 0.997. Therefore, the probability that the test signals due to organic pollutants is: \( P(O) \times P(S|O) = 0.6 \times 0.997 = 0.5982 \).

02

Calculate Probability of Signal Due to Volatile Solvents

The probability that a sample is contaminated with volatile solvents is 0.27, and the detection accuracy is 0.9995. Thus, the probability that the test signals due to volatile solvents is: \( P(V) \times P(S|V) = 0.27 \times 0.9995 = 0.269865 \).

03

Calculate Probability of Signal Due to Chlorinated Compounds

The probability that a sample is contaminated with chlorinated compounds is 0.13, and the probability of detection is 0.897. Therefore, the probability that the test signals due to chlorinated compounds is: \( P(C) \times P(S|C) = 0.13 \times 0.897 = 0.11661 \).

04

Calculate Overall Probability that the Test Signals

The test will signal if any pollutant is detected. Therefore, the overall probability is the sum of the individual probabilities: \( P(S) = P(S|O) + P(S|V) + P(S|C) = 0.5982 + 0.269865 + 0.11661 = 0.984675 \).

05

Calculate Probability of Presence of Chlorinated Compounds Given Signal

Using Bayes' theorem, the probability that chlorinated compounds are present given that the test signals is: \[ P(C|S) = \frac{P(S|C) \times P(C)}{P(S)} = \frac{0.11661}{0.984675} = 0.1184 \].

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Key Concepts

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

Bayes' Theorem

Bayes' Theorem is a very important concept in probability and statistics. It helps us understand the likelihood of an event based on prior knowledge related to that event. In simpler terms, it's about revising probabilities by incorporating new evidence.
This theorem can be very useful if you have a test that signals the presence of substances and want to determine the probability that a particular substance is causing the alarm.

• Bayes' Theorem is given by the formula \[ P(A|B) = \frac{P(B|A) \times P(A)}{P(B)} \]
  where \( P(A|B) \) is the probability of event A occurring given that B is true.
• It applies to environmental testing when we want to understand what caused a test signal.
  For example, knowing that chlorinated compounds are detected by 89.7% accuracy, we can revise the probability of their presence when a test signals.

By looking at the prior probability of contamination and detection rates, Bayes' Theorem allows for precise adjustments of probabilities.

Pollutant Detection

Pollutant detection is crucial in maintaining environmental standards and public health. The process involves using analytical methods to identify harmful substances in the environment, such as in water samples.
Each pollutant presents unique challenges and might require different detection methods.

• In this context, the exercise considers three types of pollutants: organic pollutants, volatile solvents, and chlorinated compounds.
• Methods need high sensitivity and specificity to accurately detect small amounts often harmful to human health.

The exercise emphasizes the importance of a test that can identify multiple pollutants simultaneously with varying levels of accuracy. This capacity not only improves efficiency but also reduces costs and time for testing.
Successfully detecting these pollutants ensures water quality, essential for well-being and preventing hazardous exposure.

Signal Probability

Signal probability refers to the likelihood that a test—when applied to a sample—indicates the presence of a pollutant. This can arise due to any type of contaminant detected.
The calculations in the exercise highlight actions taken to understand which pollutant might cause the system to signal.

• For each type of pollutant, compute the probability that a sample containing it leads to a test signal.
• The individual probability for organic pollutants, volatile solvents, and chlorinated compounds were calculated.

This allows researchers to calculate an overall signal probability by summing these individual probabilities. In this exercise, it resulted in an overall probability of the test signaling at approximately 98.47%.
Understanding signal probability aids in formulating the robustness of environmental tests.

Analytical Methods in Chemistry

Analytical chemistry is the study of providing information about the composition and structure of matter. It employs a variety of methods to identify and quantify substances.
These methods are essential in industries like pharmaceuticals, food safety, and environmental science. They ensure that substances meet the required safety and efficacy standards.

• The exercise focuses on an analytical method for pollutant detection in water, which can discern different types of pollutants within a single test.
• Usually, sophisticated techniques are necessary, often involving instruments such as gas chromatographs or mass spectrometers, which increase accuracy and reduce detection limits.

The novelty in the discussed exercise method is that it attempts to be versatile and comprehensive, allowing for a broader spectrum of pollutants to be detected with high accuracy. It highlights the drive for innovation and efficiency in analytical chemistry, posing that a multi-purpose and efficient test can vastly improve environmental monitoring.