After n coin-flip steps, where do you end up?
This is a one-dimensional random walk that moves +1 up or -1 down on every step. Enter the step count n, the probability p of an up-step and a target position, and the tool evaluates P(position = 2k - n) = C(n, k) x p^k x (1-p)^(n-k), along with the up-steps needed, the chance of returning to the origin and a table of nearby positions. Binomial coefficients are accumulated in log space, so up to 10,000 steps compute without overflow.
Two assumptions apply: every step is independent, and p never changes. The expected position of 0, the standard deviation sqrt(n) and the mean absolute distance sqrt(2n/pi) are reported only for the symmetric case of a 50% up-probability; with any other value those rows say so. Position and step count always share the same parity, so positions of the opposite parity have probability zero.
This page runs no repeated simulation. Averaging many random trials and the distribution of the maximum excursion are not covered. Browser random numbers are not cryptographically secure either, so never use them for legally binding or money-related official draws. Reference and study use as of October 2026.
Frequently Asked Questions
No. This page rolls no random numbers; it evaluates the distribution formula directly. The same inputs always give the same figures, and averages or standard deviations of repeated trials are not covered.
Each step changes the position by exactly 1, so position and step count always share the same parity. With an odd step count the position is odd and can never be 0, and every position of the opposite parity has probability zero for the same reason.
The sqrt(2n/pi) expression holds when the number of steps is large. For few steps the error grows, so treat it as a rough figure and read the per-position table when you need exact values.