## Fish in a Pond

There are 5 fish in a pond. What is the probability that you can split the pond into 2 halves using a diameter, so that all fish end up in one half?

**SOLUTION**

Let us generalize the problem to N fish in a pond. We can assume that all fish are on the boundary of the pond, which is a circle, and we need to find the probability that all of them are contained within a semi-circle.

For every fish **Fᵢ**, consider the semi-circle **Cᵢ** whose left end-point is at **Fᵢ**. The probability that all fish belong to **Cᵢ** is equal to **1/2ᴺ⁻¹**. Since it is impossible to have 2 fish **Fᵢ** and **Fⱼ**, such that the semi-sircles **Cᵢ** and **Cⱼ** contain all fish, we see that the probability that all fish belong to **Cᵢ** for some **i** is equal to **N/2ᴺ⁻¹**.

When N = 5, we get that the answer is 5/16.