A walker starts at 0 on the integer line and makes 4 steps. Each step is +1 or −1 with probability 1/2, independently. What is the probability that the walker is at 0 after the 4 steps?

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A walker starts at $0$ on the integer line and makes $4$ steps. Each step is $+ 1$ or $- 1$ with probability $\frac{1}{2}$ , independently. What is the probability that the walker is at $0$ after the $4$ steps?

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Hint

Count the step sequences with equally many up and down steps.

Solution

There are $16$ equally likely sequences of steps.
To end at $0$ the walker needs exactly $2$ up steps and $2$ down steps.
The number of such sequences is $(2 4) = 6$ .
The probability is $\frac{6}{16}$ , which is $\frac{3}{8}$ in lowest terms.

Answer: $\frac{3}{8}$

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