A market is in a good or a bad regime, each with chance 1/2. Given the regime, each signal is independently strong with chance 3/4 in the good regime and 1/4 in the bad one. A strong signal wins each trade with chance 3/4 and a weak one with chance 1/4, independently. Signal X wins both of its first 2 trades. What is the chance that a different signal, Y, wins its first trade? Give a fraction.

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A record that moves another signal

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A market is in a good or a bad regime, each with chance $\frac{1}{2}$ . Given the regime, each signal is independently strong with chance $\frac{3}{4}$ in the good regime and $\frac{1}{4}$ in the bad one. A strong signal wins each trade with chance $\frac{3}{4}$ and a weak one with chance $\frac{1}{4}$ , independently. Signal X wins both of its first $2$ trades. What is the chance that a different signal, Y, wins its first trade? Give a fraction.

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Hint

Update the regime first, then average over it the chance that a fresh signal wins.

Solution

In the good regime the record has chance $\frac{3}{4} \times (\frac{3}{4})^{2} + \frac{1}{4} \times (\frac{1}{4})^{2} = \frac{7}{16}$ .
In the bad regime it has chance $\frac{1}{4} \times (\frac{3}{4})^{2} + \frac{3}{4} \times (\frac{1}{4})^{2} = \frac{3}{16}$ .
So the good regime has posterior chance $\frac{7}{7 + 3} = \frac{7}{10}$ .
A fresh signal wins with chance $\frac{3}{4} \times \frac{3}{4} + \frac{1}{4} \times \frac{1}{4} = \frac{5}{8}$ in the good regime and $\frac{1}{4} \times \frac{3}{4} + \frac{3}{4} \times \frac{1}{4} = \frac{3}{8}$ in the bad one.
The answer is $\frac{7}{10} \times \frac{5}{8} + \frac{3}{10} \times \frac{3}{8} = \frac{11}{20}$ .

Answer: $\frac{11}{20}$

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