Full body scans are becoming popular even for people who have no symptoms. At any given time, a person's risk of having lung cancer is approximately 1 in 1000. If a full body scan can detect lung cancer 95% of the time (i.e., gives a positive result when a person has lung cancer) but returns a positive result 1% of the time if a person does not have lung cancer, what is the probability that a person who gets a positive test result truly has lung cancer?

Question

Quizplus · Accepted Answer

This is a classic example of applying Bayes' theorem. The probability that a person who gets a positive result truly has lung cancer is calculated as follows: $$ P(\text{Cancer}|\text{Positive}) = \frac{P(\text{Positive}|\text{Cancer}) \times P(\text{Cancer})}{P(\text{Positive})} $$Given:- $ P(\text{Cancer}) = \frac{1}{1000} $ (prior probability of having cancer)- $ P(\text{Positive}|\text{Cancer}) = 0.95 $ (sensitivity or true positive rate)- $ P(\text{Positive}|\text{No Cancer}) = 0.01 $ (false positive rate)We need to find $ P(\text{Positive}) $, the overall probability of a positive test, which is:$$ P(\text{Positive}) = P(\text{Positive}|\text{Cancer}) \times P(\text{Cancer}) + P(\text{Positive}|\text{No Cancer}) \times P(\text{No Cancer}) $$$$ P(\text{Positive}) = 0.95 \times \frac{1}{1000} + 0.01 \times \frac{999}{1000} $$$$ P(\text{Positive}) = 0.00095 + 0.00999 = 0.01094 $$Then, the probability that a person who gets a positive result truly has lung cancer is:$$ P(\text{Cancer}|\text{Positive}) = \frac{0.95 \times \frac{1}{1000}}{0.01094} \approx 0.087 $$Therefore, the correct answer is approximately 0.087, which corresponds to choice C.

Full Body Scans Are Becoming Popular Even for People Who