Exercise 1: Redoing the Base Rate Math With a Rarer Disease — Possible Solution
==================================================================================

Out of 10,000 people, 1 in 10,000 have the disease, so exactly 1
person actually has it. The remaining 9,999 people are healthy.

With a 5% false positive rate applied to those 9,999 healthy people:
9,999 x 0.05 = approximately 500 false positives (499.95, rounding to
500 for a whole number of people).

Total positive results = 1 true positive + 500 false positives = 501
total positive results.

Percentage of positives that are real (true positive rate among
positives) = 1 / 501 = approximately 0.2%.

ANSWER: Out of 10,000 people, there would be approximately 501 total
positive results (1 true positive + about 500 false positives), and
only about 0.2% of those positive results would actually reflect a
real case of the disease - an even smaller real percentage than the
chapter's own 1-in-1,000 example, because the rarer the underlying
disease, the more the false positives from an imperfect test
dominate the pool of positive results.

WHY THIS WORKS AS AN ANSWER
------------------------------
This correctly redoes the chapter's own real calculation with the new
numbers, showing the same step-by-step reasoning (true positives vs.
false positives among the healthy population) applied to a rarer
condition.