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Profit per day of a repeating trade

ProbabilityMedium

A convergence trade is opened, closed and reopened at once, in independent cycles. With probability $0.75$ a cycle lasts $2$ days and earns $6$ . Otherwise it lasts $8$ days and loses $2$ . What is the long-run profit per day? Give a fraction in lowest terms.

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Hint

Divide expected profit per cycle by expected length of a cycle.

Solution

Expected profit per cycle: $0.75 \times 6 - 0.25 \times 2 = 4$ .
Expected length: $0.75 \times 2 + 0.25 \times 8 = 3.5$ .
The ratio gives the long-run rate: $\frac{4}{3.5} = \frac{8}{7}$ .

Answer: $\frac{8}{7}$

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