Random Walk Probability Calculator

How it is calculated

For a walk of n steps, each step goes right (+1) with probability p and left (−1) with probability q = 1 − p. To finish at net position k you need r = (n + k) / 2 right steps, and then P = C(n, r) × pr × qn−r. The tool also reports r and the expected final position (2p − 1) × n. Enter p as a fraction between 0 and 1 (0.6, not 60). If n + k is odd, or |k| is larger than n, the position cannot be reached and the probability is shown as 0.000000. Steps and position are rounded to whole numbers, and a negative n or a p outside 0 to 1 gives no result. The probability is shown to six decimals.

Common mistakes

• Typing p as a percentage (60) instead of a fraction (0.6). Values above 1 are rejected and the tool returns no result.
• Picking an impossible target. After 10 steps you can only be at an even position, so k = 3 gives 0.000000. This is correct, not an error: the position always has the same parity as n.
• Confusing k with the number of right steps. k is the net displacement (right minus left); the right steps r = (n + k) / 2 are shown separately.
• Reading the result as the probability of reaching k at some point during the walk. It is the probability of being exactly at k after the last step, not of ever visiting it.
• Comparing the expected position with the most likely one. The expected value (2p − 1) × n is an average and may not be an attainable position itself, for example with odd n and p = 0.5.