For a particular population group it is estimated that the prevalence of prostate cancer is 15.2%.
A new diagnostic procedure for this disease has a sensitivity of 0.934, correctly diagnosing an
individual with prostate cancer 93.4% of the time. The specificity of the procedure is 0.898, so that
an individual without the disease is correctly diagnosed 89.8% of the time.
Suppose the diagnostic procedure indicates an individual from this population group has the disease. What is probability that they actually do have prostate cancer?

To find the probability that an individual actually has prostate cancer given that the diagnostic procedure indicates they have the disease, we use Bayes' theorem. This theorem relates the conditional and marginal probabilities of random events.

Given:
- Prevalence of prostate cancer, \( P(C) = 0.152 \)
- Sensitivity of the diagnostic procedure, \( P(T^+ | C) = 0.934 \)
- Specificity of the diagnostic procedure, \( P(T^- | C^-) = 0.898 \)

We need to determine \( P(C | T^+) \), the probability that an individual has prostate cancer given a positive test result.

First, we find the necessary complementary probabilities:
- Probability of not having prostate cancer, \( P(C^-) = 1 - P(C) = 1 - 0.152 = 0.848 \)
- Probability of a positive test given no prostate cancer, \( P(T^+ | C^-) = 1 - P(T^- | C^-) = 1 - 0.898 = 0.102 \)

Bayes' theorem states:
\[ P(C | T^+) = \frac{P(T^+ | C) \cdot P(C)}{P(T^+)} \]

To find \( P(T^+) \), the overall probability of a positive test result, we use the law of total probability:
\[ P(T^+) = P(T^+ | C) \cdot P(C) + P(T^+ | C^-) \cdot P(C^-) \]

Plugging in the values:
\[ P(T^+) = (0.934 \cdot 0.152) + (0.102 \cdot 0.848) \]
\[ P(T^+) = 0.141968 + 0.086496 \]
\[ P(T^+) = 0.228464 \]

Now we can apply Bayes' theorem:
\[ P(C | T^+) = \frac{0.934 \cdot 0.152}{0.228464} \]
\[ P(C | T^+) = \frac{0.141968}{0.228464} \]
\[ P(C | T^+) \approx 0.6216 \]

Therefore, the probability that an individual actually has prostate cancer given that the diagnostic procedure indicates they have the disease is approximately 62.16%.

Would you like more details or have any questions?

Here are some related questions:
1. What is the impact of sensitivity on the positive predictive value (PPV) of a test?
2. How does specificity influence the false positive rate of a diagnostic procedure?
3. What is the difference between prevalence and incidence in medical statistics?
4. How can we improve the diagnostic accuracy of a medical test?
5. What are the consequences of a high false negative rate in cancer screening?
6. What role does Bayes' theorem play in medical diagnostics?
7. How does the prevalence of a disease affect the predictive values of a diagnostic test?
8. Can you provide an example where a high prevalence of a disease significantly changes the predictive values?

**Tip:** When interpreting diagnostic test results, always consider the prevalence of the disease in the population as it greatly affects the positive and negative predictive values.

Learn how to calculate the probability that an individual actually has prostate cancer given a positive diagnostic test result. This problem involves understanding probability theory, Bayes' theorem, and conditional probability. Suitable for undergraduate students and professionals in medical diagnostics.

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