Decisions
half a day

Monte Carlo simulation

Sample each uncertain input many times and record the distribution of results, rather than computing one answer from best guesses.

Time cost
half a day
Output
A distribution with percentiles and a threshold probability.
Steps
5

Use when

  • Several uncertain inputs combine and you need the shape of the result, not its average.
  • You care about the probability of missing a threshold rather than the central estimate.
  • Inputs interact non-linearly, so plugging in averages gives the wrong answer.

Do not use when

  • There are one or two uncertain inputs. A range and a sensitivity check is faster and clearer.
  • The input distributions are invented. Sampling from fiction produces a precisely shaped fiction.

Inputs required

  • A model connecting inputs to outputs
  • A distribution per uncertain input
  • A few thousand iterations

Procedure

  1. 01

    Write the model

    The arithmetic connecting inputs to the output you care about. It must run end to end on a single set of inputs first.

  2. 02

    Give each input a distribution

    Not a point. A triangular distribution — low, likely, high — is usually enough and is easy to elicit honestly.

  3. 03

    Handle correlation

    Decide which inputs move together. Sampling correlated inputs independently is the most common error and it understates the tails badly.

  4. 04

    Run and collect

    A few thousand iterations. Keep every result, not the mean.

  5. 05

    Read the distribution, not the average

    Report the 10th, 50th and 90th percentiles and the probability of crossing whatever threshold matters. The average is often the least useful number in the output.

Characteristic failure mode

Precision theatre. A smooth curve from 10,000 iterations looks like measurement, but its shape is entirely determined by input distributions that were guessed. The simulation cannot be more accurate than its inputs.

Worked example

Estimating whether a budget of £250k covers a project with four uncertain cost lines.

  1. 01Each line gets a low/likely/high triangular range.
  2. 02Two lines are correlated — both depend on the same supplier — and are sampled together.
  3. 035,000 runs.

Result

Median £228k, 90th percentile £291k, and a 27% chance of exceeding £250k. Plugging in the four “likely” values alone would have given £221k and a false sense of comfort.

Where to go next