# Gambler's Luck

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 $$\frac{a}{b}$$.  What is $$b-a$$?  $$\textbf{Details and Assumptions:}$$

• 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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