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Chance of a profitable book

Market makingHard

A market maker sells $6$ binary contracts at $0.7$ each. Each contract pays $1$ with probability $0.5$ and $0$ otherwise, independently of the others, and the market maker pays out on settlement. What is the probability that the total profit is strictly positive? Give an exact fraction.

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Hint

Profit is total premium minus the number of contracts that pay.

Solution

The premium received is $6 \times 0.7 = 4.2$ .
The profit is positive when the number of paying contracts is at most $4$ .
The probability is $(1 + 6 + 15 + 20 + 15) \div 64 = \frac{57}{64}$ .

Answer: $\frac{57}{64}$

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