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Hitting Time Calculator

Enter the boundary, start and step probability to find the expected hitting time.

How it is calculated

The walk lives on positions 0 to N and moves +1 with probability p and −1 with probability q = 1 − p each step. The result is the expected number of steps until it first touches 0 or N. For a fair walk (p = 0.5) the formula is E = i × (N − i). Otherwise, with r = q / p: E = i / (q − p) − N / (q − p) × (1 − ri) / (1 − rN). Here N is the boundary (states), i the start position and p the up-step probability. Inputs are rejected if N ≤ 0, i is outside 0…N, or p is not strictly between 0 and 1. N and i must be whole numbers. Starting on a boundary gives 0 steps. When p is very close to 0.5 the tool solves the step equations directly instead of using the closed form, so the result stays accurate and approaches i × (N − i). The answer is an average over many walks, not the length of any single one.

Worked example

Take N = 10, start i = 3 and p = 0.6, so q = 0.4. Then r = 0.4 / 0.6 = 0.6667 and q − p = −0.2. The first term is 3 / (−0.2) = −15. The second uses N / (q − p) = −50 and (1 − 0.6667³) / (1 − 0.6667¹⁰) = 0.7037 / 0.9827 = 0.7161. So E = −15 − (−50 × 0.7161) = −15 + 35.81 = 20.81 steps on average. With a fair coin (p = 0.5) the same start would give 3 × 7 = 21.

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Common mistakes

  • Treating p as a percentage. Enter 0.6, not 60; a value of 1 or more is rejected, as is 0.
  • Reading the result as the probability of hitting 0 rather than N. The tool gives only the expected time to hit either boundary, not which one.
  • Starting position outside the range. The start must lie between 0 and N inclusive, and the walk is only defined for whole-number positions, so use integers.
  • Expecting a biased walk to take much longer. A strong drift towards one end usually makes the time shorter, because the walk is pushed to a boundary.
  • Reading the "states" field as the count of positions. It is the upper boundary N: the walk has the N + 1 positions 0 to N, so N = 10 means positions 0 to 10. Non-whole numbers for N or the start give no result.

About this calculator

In a one-dimensional random walk confined between two boundaries — 0 and N — the expected hitting time answers a classic probability question: starting from some position between the boundaries, how many steps, on average, does it take to first reach one of the boundaries? This is a version of the famous gambler's ruin problem, and the expected time depends on the starting position, the distance to each boundary, and the probability of stepping in each direction at every step.

This calculator takes the boundary values, a starting position, and the step probability, then computes the expected hitting time using the closed-form solution to the corresponding difference equation. Probability theory students working through Markov chain problem sets, and quantitative analysts modeling barrier-crossing scenarios in risk or options pricing use it to verify hand calculations or to explore how the expected time changes as the starting position or step bias shifts.

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