Two steps, an even chance of returning

Question

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

Accepted Answer

The four equally likely step pairs are up-up, up-down, down-up, down-down. Only up-down and down-up cancel out to return to 0, which is 2 of the 4 cases: \frac{2}{4} = \frac12. Equivalently, returning to 0 needs one up and one down, and \binom{2}{1} = 2 orderings out of 2^2 = 4. Takeaway. After two fair steps, returning to the start is a coin flip: exactly 1/2.

Two steps, an even chance of returning

Your answer

More Probability questions