Assessing the evidence differently
Alice is randomly selected from her population to be tested for a rare genetic disorder that affects about one in 10,000 people. The test is accurate 99 percent of the time, both among subjects that have the disorder and among subjects that don't. Alice's test comes back positive.
Call the information in the previous paragraph E, and suppose it's all you know about the situation. How confident are you that Alice has the disorder?
Letting our subjective probabilities be guided by the stated frequencies, we can use Bayes' Theorem to figure out that P(disorder | positive) = P(positive | disorder) * P(disorder) / (P(positive | disorder) * P(disorder) + P(positive | ~disorder) * P(~disorder)) = 0.99 * 0.0001 / (0.99 * 0.0001 + 0.01 * 0.9999) = 0.0098. Assume then that your degree of belief is about 0.01.