Lying Fairy

Logic Level 3

There are 100 islands, each of a distinct size. 100 perfectly logical villagers live on each island, and the numbers of villagers with blue eyes on each of the islands are the distinct numbers \(0,1,\ldots,99\), with a uniform distribution.

On one of the islands, all the villagers know that everyone has either blue or brown eyes, but it is forbidden for a villager to know his/her own eye color. If a villager believes they know their own eye color on a given day, they must leave the village in exile that night.

One day a fairy visits this island and announces that at least 50 villagers have blue eyes. Unfortunately, the fairy chooses completely randomly whether to lie or to the truth, but each villager believes the fairy unless they observe a contradiction. The villagers do not discuss the fairy's statement.

What is the probability that a given random villager from the smallest island will eventually leave the island in exile but be incorrect about their eye color?

If the probability is \(\frac{p}{q}\), where \(p,q\) are positive, coprime integers, give your answer as \(p+q\).


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