Final answer:
In a diagnostic test that is 90% accurate for a disease with a prevalence rate of 1 in 1000, the probability that a person has the disease after testing positive is approximately 0.89%. This surprising result is due to the disease's rarity and the relative abundance of false positives generated by the test.
Step-by-step explanation:
The switched-conditionals fallacy refers to a common error in interpreting the results of diagnostic tests where the conditionality between the probability of a positive test given the presence of disease and the probability of the disease given a positive test is mistakenly reversed. To calculate the chances that a person actually has the disease after testing positive in this scenario, we use Bayes' theorem.
Let's calculate the probability:
- Probability of having disease (D) = 1/1000 = 0.001.
- Probability of not having disease (not D) = 1 - 0.001 = 0.999.
- Probability of testing positive given disease (True Positive, TP) = 0.9.
- Probability of testing positive given no disease (False Positive, FP) = 0.1.
- Probability of a random person testing positive (P) = (TP * Probability of D) + (FP * Probability of not D).
- P(TP|D) * P(D) = 0.9 * 0.001 = 0.0009.
- P(FP|not D) * P(not D) = 0.1 * 0.999 = 0.0999.
- P = 0.0009 + 0.0999 = 0.1008.
- So, the probability that a person actually has the disease given they've tested positive (P(D|TP)) is P(TP|D) * P(D) / P = 0.0009 / 0.1008.
Finally, P(D|TP) = 0.0089 or 0.89%.
Therefore, despite the high accuracy rate of the test, the probability that a person actually has the disease given that they tested positive is only about 0.89%, which is still quite low. This is a result of the disease being rare and the proportion of false positives (due to the 10% inaccuracy of the test) being high in comparison to the true positives.