Four in a row the same way

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 +4?

Accepted Answer

Ending at +4 after 4 steps means every step must be +1; that is the only path that reaches the far end. Each step is up with probability 1/2, and the steps are independent, so \left(	frac12ight)^4 = \frac{1}{16}. Takeaway. Reaching the extreme endpoint takes one exact sequence, so the probability is (1/2)^n; for 4 steps that is 1/16.

Four in a row the same way

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