Problems Involving Probability, Bayes' Theorem, And Conditio
Problems involving probability, Bayes theorem, and conditional probability
A bin is filled with half red balls and half purple balls. Approximately 20% of the red balls have a stripe on them, and the others do not. Approximately 90% of the purple balls have a stripe on them and the others do not. A ball is randomly selected, and it is striped. What is the probability it is purple? (Hint: Draw a tree diagram)
Suppose that the reliability of a test for hepatitis is specified as follows: Of people with hepatitis, 95% have a positive reaction and 5% have a negative reaction; of people free of hepatitis, 90% have a negative reaction. From a large population of which 0.05% of the people have hepatitis, a person is selected at random and given the test. If the test is negative, what is the probability that the person actually has hepatitis?
Use Bayes' theorem. There are three sections of English 101. In section I, there are 25 students, of whom 5 are mathematics majors. In section II, there are 20 students, of whom 6 are mathematics majors. In section III, there are 35 students, of whom 5 are mathematics majors. A student in English 101 is randomly selected. Find the probability that the student is from section I, given that he or she is a mathematics major.
Paper For Above instruction
The given problems involve fundamental concepts of probability theory and Bayesian inference, which are critical tools in statistical reasoning and decision-making under uncertainty. These problems exemplify the application of conditional probability, Bayes' theorem, and basic probability rules, which are essential for understanding the likelihood of events in diverse real-world contexts such as quality testing, medical diagnosis, and educational data analysis.
Problem 1: Conditional probability with a bin containing red and purple balls
In the first scenario, a bin contains an equal number of red and purple balls, representing a classical case of categorical probability. The problem specifies that among the red balls, 20% possess stripes, while the remaining 80% do not. Conversely, 90% of the purple balls have stripes. When a ball is randomly drawn from the bin, and it is observed to be striped, the question asks for the probability that this striped ball is purple.
This problem can be approached with Bayes' theorem, which states:
P(Purple | Striped) = (P(Striped | Purple) * P(Purple)) / P(Striped)
Where:
- P(Purple) = 0.5 (since half the balls are purple)
- P(Red) = 0.5
- P(Striped | Purple) = 0.9
- P(Striped | Red) = 0.2
Calculating P(Striped):
P(Striped) = P(Striped | Purple) P(Purple) + P(Striped | Red) P(Red) = (0.9)(0.5) + (0.2)(0.5) = 0.45 + 0.1 = 0.55
Applying Bayes' theorem:
P(Purple | Striped) = (0.9)(0.5) / 0.55 ≈ 0.45 / 0.55 ≈ 0.8182
Therefore, given that the selected ball is striped, there is approximately an 81.82% chance it is purple.
Problem 2: Medical diagnosis using Bayes' theorem
The second problem involves assessing the probability of having hepatitis given a negative test result, considering the test's sensitivity and specificity and the prevalence in the population.
Given data:
- P(Heptatitis) = 0.0005 (0.05%)
- P(Healthy) = 0.9995
- P(Negative | Heptatitis) = 0.05 (5%) false negatives)
- P(Negative | Healthy) = 0.9 (90% true negatives)
We are asked to find P(Heptatitis | Negative), the probability that a person has hepatitis given a negative test, which can be calculated using Bayes' theorem:
P(Heptatitis | Negative) = [P(Negative | Heptatitis) * P(Heptatitis)] / P(Negative)
Where P(Negative) is the total probability of testing negative:
P(Negative) = P(Negative | Heptatitis) P(Heptatitis) + P(Negative | Healthy) P(Healthy) = (0.05)(0.0005) + (0.9)(0.9995) ≈ 0.000025 + 0.89955 ≈ 0.899575
Thus,
P(Heptatitis | Negative) ≈ (0.05)(0.0005) / 0.899575 ≈ 0.000025 / 0.899575 ≈ 2.78 × 10-5
Interpreting this result, the probability that a person with a negative hepatitis test actually has hepatitis is approximately 0.00278%, highlighting the importance of test accuracy and the low prevalence of hepatitis in the population.
Problem 3: Conditional probability in classroom sections
The third problem involves applying Bayes' theorem to a real-world educational setting, calculating the probability that a randomly chosen mathematics major is from Section I, given the total student distribution across three sections.
Data:
- Section I: 25 students, 5 are math majors
- Section II: 20 students, 6 are math majors
- Section III: 35 students, 5 are math majors
Total students: 25 + 20 + 35 = 80
Total math majors: 5 + 6 + 5 = 16
The probability that a student is from Section I given that they are a math major is:
P(Section I | Math Major) = P(Math Major | Section I) * P(Section I) / P(Math Major)
Where:
- P(Section I) = 25/80 = 0.3125
- P(Section II) = 20/80 = 0.25
- P(Section III) = 35/80 = 0.4375
- P(Math Major | Section I) = 5/25 = 0.2
- P(Math Major | Section II) = 6/20 = 0.3
- P(Math Major | Section III) = 5/35 ≈ 0.1429
Calculating P(Math Major):
P(Math Major) = (P(Section I) P(Math Major | Section I)) + (P(Section II) P(Math Major | Section II)) + (P(Section III) * P(Math Major | Section III))
= (0.3125)(0.2) + (0.25)(0.3) + (0.4375)(0.1429) ≈ 0.0625 + 0.075 + 0.0625 = 0.2
Finally, applying Bayes' theorem:
P(Section I | Math Major) = (0.2)(0.3125) / 0.2 = 0.0625 / 0.2 = 0.3125
Therefore, given that a student is a mathematics major, there is a 31.25% probability that the student is from Section I.
Conclusion
These problems demonstrate the practical application of probability concepts and Bayesian reasoning in different contexts. The first problem illustrates how prior probabilities and conditional probabilities combine to update beliefs about an event. The second emphasizes the importance of understanding test accuracy and disease prevalence in medical diagnosis. The third showcases how Bayesian inference applies to categorical data in educational settings, aiding in understanding student distribution and characteristics.
References
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