In a winner-takes-all bet, a gambler is given \(2\) coins: one \(\text{fair}\), and the other \(\text{unfair}\). The \(\text{fair}\) coin comes up \(\text{Heads}\) \(\text{50%}\) of the time, and the \(\text{unfair}\) coin comes up \(\text{Tails}\) \(\text{10%}\) of the time. The gambler is asked to identify the \(\text{unfair}\) coin. Assuming he plays optimally, the probability that he does not identify the unfair coin is \(\frac{a}{b}\).
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What is \(b-a\)?
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\(\textbf{Details and Assumptions:}\)

The \(2\) coins are otherwise identical.

The gambler is allowed \(\text{only } 2\) tosses to determine which coin is unfair.

The gambler is allowed to choose \(\text{either of the coins }\)for the first toss. Subsequently, he is again allowed to choose \(\text{either of the coins }\)for the second toss.

\(a\) and \(b\) are co-prime.

This is part of Ordered Disorder.\[\] Please do see Gambler's Luck Version 2.\[\] The problem was inspired by Matt Enlow.

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