You are given a bag of 100 coins, with 99 fair ones flipping heads and tails with 0.50 probability each, and one unfair coin which flips heads with 1.0 probability. You pick a coin randomly and flip it 10 times, getting heads every single time. What is the probability that you picked the unfair coin?
The question can be answered quite easily using Bayes' Theorem from probability theory.