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Hitting the inventory limit

Market makingMedium

A market maker starts flat and each trade changes its inventory by $+ 1$ or $- 1$ with equal probability, independently. Its long limit is $+ 3$ . What is the probability that inventory reaches $+ 3$ at some point during the first $7$ trades? Give a fraction in lowest terms.

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Hint

Use reflection: count the paths by where they end.

Solution

Out of $2^{7} = 128$ paths, those ending at $3$ , $5$ and $7$ number $(5 7) = 21$ , $(6 7) = 7$ and $(7 7) = 1$ .
Reflection adds the paths that touch $+ 3$ and end lower. They pair with paths ending above $3$ , of which there are $7 + 1 = 8$ .
Total touching paths: $21 + 7 + 1 + 8 = 37$ , so the chance is $\frac{37}{128}$ .

Answer: $\frac{37}{128}$

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