Probability Concepts During An Epidemic Of Disease, A Doctor

Probability Concepts During an epidemic of disease, a doctor sees 110 people who have symptoms commonly associated with the disease

During an epidemic of disease, a doctor examines 110 patients presenting symptoms commonly associated with a particular illness. Of these, 45 are women, with 20 women actually having the disease. Among the men, 15 have the disease. This scenario provides an opportunity to explore various probability concepts, including conditional probability, joint probability, and multiple event outcomes. The problem involves defining relevant events, visualizing their relationships through Venn diagrams, calculating associated probabilities, and analyzing test accuracy in diagnosing the disease.

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The epidemic scenario described involves crucial probability concepts that are fundamental in public health decision-making and diagnostic testing. By dissecting this data, we can better understand the relationships among gender, disease status, and diagnostic outcomes. The following analysis elaborates on these concepts through models, calculations, and interpretations.

Part (a): Drawing a Venn diagram

To visualize the problem, consider two events: W (the person is a woman) and D (the person has the disease). The total number of symptomatic individuals is 110, with 45 women and 65 men. Among these, 20 women and 15 men have the disease. The remaining symptomatic individuals do not have the disease.

In the Venn diagram, the circle W represents all women with symptoms (45 individuals), and within this circle, the subset D ∩ W (women with the disease) contains 20 individuals. Outside this subset but within W are women without the disease (25 individuals). Similarly, the circle D represents all individuals with the disease. Inside D, the subset D ∩ W accounts for 20 women with the disease, and D ∩ M (total men with disease) is 15, meaning 15 men have the disease. The remaining men who have symptoms but not disease are 50 individuals (65 total men minus 15 with disease). The total non-diseased individuals are therefore 110 minus the 35 with disease, which equals 75. These are made up of women without the disease (25), men without disease (50), totaling 75.

Part (b): Descriptive probabilities and calculations

Let's define the events clearly:

  • W: the selected person is a woman
  • W^c: the selected person is a man (complement of W)
  • D: the person has the disease
  • W ∩ D: the person is a woman with the disease (20)
  • W ∩ D^c: woman without the disease (25)
  • W^c ∩ D: man with the disease (15)
  • W^c ∩ D^c: man without the disease (50)

Calculations of probabilities:

  • P(W) = 45 / 110 ≈ 0.409
  • P(W^c) = 65 / 110 ≈ 0.591
  • P(D) = (20 + 15) / 110 = 35 / 110 ≈ 0.318
  • P(D | W) = (20) / 45 ≈ 0.444
  • P(D | W^c) = (15) / 65 ≈ 0.231
  • P(W ∩ D) = 20 / 110 ≈ 0.182
  • P(W ∩ D^c) = 25 / 110 ≈ 0.227
  • P(W^c ∩ D) = 15 / 110 ≈ 0.136
  • P(W^c ∩ D^c) = 50 / 110 ≈ 0.455

Part (c): Probability for three randomly selected individuals

i) Probability that all three individuals have the disease:

P(each has disease) = P(D) = 0.318


Since the selections are independent, the probability that all three are diseased:


P(all three diseased) = P(D)^3 ≈ (0.318)^3 ≈ 0.0322

ii) Probability that exactly one of the three has the disease:

Number of ways to choose one diseased individual out of three:


C(3,1) = 3


Probability that one is diseased and two are not:


P = 3 × P(D) × (1 - P(D))^2


≈ 3 × 0.318 × (0.682)^2


≈ 3 × 0.318 × 0.465


≈ 3 × 0.148


≈ 0.444

Part (d): Diagnostic test accuracy and conditional probabilities

The diagnostic test's sensitivity (true positive rate): 95%, meaning a person with the disease tests positive with probability 0.95. The specificity (true negative rate): 92%, since 8% of people without the disease test positive (false positives).

i) Probability that a randomly selected person reacts positively:

P(positive) = P(positive | D) × P(D) + P(positive | D^c) × P(D^c)


= (0.95) × (0.318) + (0.08) × (1 - 0.318)


= 0.95 × 0.318 + 0.08 × 0.682


≈ 0.302 + 0.055


≈ 0.357

ii) Probability that a person has the disease given that they reacted positively (posterior probability, or positive predictive value):

Using Bayes' theorem:


P(D | positive) = [P(positive | D) × P(D)] / P(positive)


= 0.95 × 0.318 / 0.357


≈ 0.302 / 0.357


≈ 0.846

Conclusion

This analysis illustrates the application of probability concepts to an epidemic scenario, highlighting how gender and disease status interplay and how diagnostic test characteristics influence disease detection. The calculations underscore the importance of sensitivity, specificity, and predictive values in interpreting test results. Such probabilistic assessments are vital for effective disease management, resource allocation, and epidemiological understanding, ultimately contributing to more informed health interventions.

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