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Profit until the first informed order

Market makingHard

Each order is informed with probability $0.1$ , independently of the others, and otherwise retail. A retail order earns the market maker $1$ and an informed order costs $10$ . The market maker keeps quoting until the first informed order arrives, and that order is included. What is the expected total profit?

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Hint

Count the expected number of retail orders before the first informed order.

Solution

The number of retail orders before the first informed order has expectation $0.9 \div 0.1 = 9$ .
Those orders earn $9 \times 1 = 9$ in expectation, and the informed order costs $10$ .
The expected total profit is $9 - 10 = - 1$ .

Answer: $- 1$

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