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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