Four steps ending two ahead

Question

A marker starts at 0 and takes 4 steps. Each step is +1 or -1, each with probability 1/2, independently. What is the probability the marker ends at +2?

Accepted Answer

To finish at +2 after 4 steps you need three +1 steps and one -1 step, since 3 - 1 = 2. The number of ways to choose which one of the 4 steps is the down-step is \binom{4}{1} = 4, out of 2^4 = 16 equally likely sequences: \frac{4}{16} = \frac14. Takeaway. Count the up-and-down split that lands you on the target, then divide by 2^n: here 4/16 = 1/4.

Four steps ending two ahead

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