Three steps, all forward?

Question

A token starts at 0 and takes 3 steps. Each step is +1 or -1, each with probability 1/2, independently. What is the probability the token ends at +3?

Accepted Answer

Ending at +3 after 3 steps requires every single step to be +1; there is no other way. Each step is up with probability 1/2, and the three steps are independent, so \left(	frac12ight)^3 = \frac18. Takeaway. The farthest endpoints come from one unique path, so their probability is just (1/2)^n; here 1/8.

Three steps, all forward?

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