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Flat after a burst of flow

Market makingEasy

A market maker starts flat. Each trade changes its inventory by $+ 1$ or $- 1$ with equal probability, independently. What is the probability that it is exactly flat after $6$ trades? Give a fraction in lowest terms.

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Hint

Being flat needs equal numbers of each kind of trade.

Solution

Flat means $3$ buys and $3$ sells, in any order: $(3 6) = 20$ of the paths.
There are $2^{6} = 64$ equally likely paths, so the chance is $\frac{20}{64} = \frac{5}{16}$ .

Answer: $\frac{5}{16}$

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