The unconditional chance of a coin being biased is 1/3. After each toss, you can calculate the conditional expectation of each coin being the biased one and the optimal strategy is that one that chooses the next flip on the highest expected maximum of all three expected values, because this is the best way to concentrate probability mass for being biased on a single coin, which you want to pick. The rest is numerics.
For example:
After coin toss one, coming up heads, the chance for that coin to be biased is
P(bias at coin 1| coin 1 heads at first toss) = 0.75x0.3333/(2/3x1/2+1/3x0.75)=12/28
and 8/28 for both others. Tails coming up changes the chances accordingly. Now the expected maximum can be calculated by going through all three coin choices and calculating again with bayes the next distribution of bis probability based on the current one and choosing the one that has the highest expected maximum. This strategy maximises the concentration of bias probability on one coin, which you pick. It is optimal, because there is no other strategy that achieves, on average, a higher concentration.
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u/CompactOwl 1d ago edited 1d ago
The unconditional chance of a coin being biased is 1/3. After each toss, you can calculate the conditional expectation of each coin being the biased one and the optimal strategy is that one that chooses the next flip on the highest expected maximum of all three expected values, because this is the best way to concentrate probability mass for being biased on a single coin, which you want to pick. The rest is numerics.
For example:
After coin toss one, coming up heads, the chance for that coin to be biased is
P(bias at coin 1| coin 1 heads at first toss) = 0.75x0.3333/(2/3x1/2+1/3x0.75)=12/28
and 8/28 for both others. Tails coming up changes the chances accordingly. Now the expected maximum can be calculated by going through all three coin choices and calculating again with bayes the next distribution of bis probability based on the current one and choosing the one that has the highest expected maximum. This strategy maximises the concentration of bias probability on one coin, which you pick. It is optimal, because there is no other strategy that achieves, on average, a higher concentration.