A market maker starts flat with a limit of 2 units long or short. In each of 4 rounds one unit is traded, bought or sold with probability 1/2 each, independently, and the position changes by 1. What is the probability that the position reaches the limit in at least one round? Give an exact fraction.

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Chance of reaching the limit

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A market maker starts flat with a limit of $2$ units long or short. In each of $4$ rounds one unit is traded, bought or sold with probability $\frac{1}{2}$ each, independently, and the position changes by $1$ . What is the probability that the position reaches the limit in at least one round? Give an exact fraction.

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Hint

Count the paths that never touch the limit.

Solution

There are $2^{4} = 16$ equally likely paths.
A path avoids the limit only if it alternates between a nonzero position and $0$ , and there are $2 \times 1 \times 2 \times 1 = 4$ such paths.
The probability is $1 - \frac{4}{16} = \frac{3}{4}$ .

Answer: $\frac{3}{4}$

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