Gambler's Luck

In a winner-takes-all bet, a gambler is given 22 coins: one fair\text{fair}, and the other unfair\text{unfair}. The fair\text{fair} coin comes up Heads\text{Heads} 50%\text{50\%} of the time, and the unfair\text{unfair} coin comes up Tails\text{Tails} 10%\text{10\%} of the time. The gambler is asked to identify the unfair\text{unfair} coin. Assuming he plays optimally, the probability that he does not identify the unfair coin is ab\frac{a}{b}. What is bab-a? Details and Assumptions:\textbf{Details and Assumptions:}

  • The 22 coins are otherwise identical.

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

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

  • aa and bb 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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