HesapHanem

Random Walk Probability Calculator

Enter the number of steps, target position and step probability to get the result.

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, an n above 10,000,000 or a p outside 0 to 1 gives no result. The probability is shown to six decimals, and long walks work too: n = 2000, k = 0, p = 0.5 gives 0.017839.

Worked example

Ten steps, ending two places to the right of the start, with p = 0.6. Right steps: r = (10 + 2) / 2 = 6, so 4 steps go left. The number of orderings is C(10, 6) = 210. Then 0.66 = 0.046656 and 0.44 = 0.0256, and 210 × 0.046656 × 0.0256 is about 0.2508. The tool shows a probability of 0.250823, with 6 right steps and an expected position of 2.00.

Try this example →

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.

Sources

About this calculator

The random walk probability calculator computes, for a simple 1D random walk of n steps where each step moves +1 with probability p and −1 with probability q = 1 − p, the probability of ending at a specific position k. Since ending at position k after n steps requires exactly (n+k)/2 forward steps and (n−k)/2 backward steps, the answer follows the binomial distribution: P(k) = C(n, (n+k)/2) × p^((n+k)/2) × q^((n−k)/2).

The random walk is one of the most widely reused models in applied math: it describes the diffusion of a particle in Brownian motion, is the discrete-time skeleton behind stock-price models like geometric Brownian motion, and shows up in gambler's-ruin problems in probability theory. Statistics and finance students, and researchers modeling stochastic processes use this calculator to check the closed-form binomial result against a simulation or textbook problem.

Was this helpful?

You can pick up to 3 reactions.

Comments (1)

  • Editorial team

    A fair walk is a good way to see how fast certainty fades. The chance of landing exactly back at the start is 0.246094 after 10 steps, 0.079589 after 100 and 0.025225 after 1,000, shrinking roughly with the square root of the step count. The spread of outcomes widens as the walk lengthens, so for long walks a range of positions is usually a better question than one exact spot. What kind of walk are you modelling?

Popular calculators

Loan Calculator

Monthly payment, total interest and full amortization schedule.

Percentage Calculator

Percent change, percent of, and ratio calculations.

Discount Calculator

Calculate final price, savings, and discount amount.

BMI Calculator

Body mass index from height and weight (metric or imperial).

Age Calculator

Compute your exact age in years, months and days from a birth date.

Compound Interest Calculator

Calculate the final amount and interest earned with compounding.

All Calculators