# Heads, tails, and what else?

Level pending

In front of you are two identical-looking coins. One is a standard fair coin, but the other one is a very special kind of coin, characterized as follows:

1. There is always an equal probability of heads or tails.
2. On the $$\textrm{n}^{\textrm{th}}$$ flip, there is a $$1 - \frac{1}{n}$$ probability that the coin will land on its side!

You do not know which is which, so you pick up one of them randomly, and then you flip it 5 times.

The probability that you will be able to identify the special coin given 5 flips can be expressed as $$\frac{a}{b}$$ where $$a$$ and $$b$$ are coprime positive integers. Find $$a + b$$.

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