Mathematics
Mathematics
Problem
A blood test has been designed to detect a particular form of cancer. The probability that the test correctly identifies someone with the cancer is 0.97, and the probability that the test correctly identifies someone without the cancer is 0.93. Approximately 0.1% of the general population are known to contract this cancer.When a patient has a blood test, the test results are positive for the cancer. Find the probability that the patient actually has the cancer.Intuitive explanation
This is a classic application of Bayes' theorem, where you need to flip the conditional probability around. You're given the probability of a positive test given cancer, but you really want the probability of cancer given a positive test—a very different thing. The key insight is that even though the test is quite accurate, cancer is so rare in the population that most positive results are actually false alarms. Set up a two-way table or use Bayes' formula directly, accounting for both the true positive rate and the false positive rate weighted by their actual prevalence.
Intuitive explanation
This is a classic application of Bayes' theorem, where you need to flip the conditional probability around. You're given the probability of a positive test given cancer, but you really want the probability of cancer given a positive test—a very different thing. The key insight is that even though the test is quite accurate, cancer is so rare in the population that most positive results are actually false alarms. Set up a two-way table or use Bayes' formula directly, accounting for both the true positive rate and the false positive rate weighted by their actual prevalence.