What Monte-Carlo Does

Many trials, one spread

Business Risk AnalysisMonte-Carlo IntuitionFree preview

⏱️ About 16 min

Northwind Trading cannot name one exact cost for a build, only a range. It runs many random trials drawn from a triangular distribution with an optimistic $800,000, a most-likely $1,000,000, and a pessimistic $1,500,000. The simulated spread clusters around the mean of (800,000 + 1,000,000 + 1,500,000)/3 = $1,100,000, showing both the average and how wide the outcomes can run.

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The big idea: Monte-Carlo runs many random trials from an input distribution and shows the spread of outcomes; the average of those trials converges toward the distribution mean. For a triangular distribution with optimistic (min), most-likely (mode), and pessimistic (max) values, the theoretical mean is (min + mode + max) / 3.

🎯 By the end, you'll be able to

Describe Monte-Carlo as many random trials drawn from a distribution.
Compute the triangular mean as (min + mode + max) / 3.
Read the spread and average that the trials produce.

📎 Helpful to know first

Completion of Module 4: Expected Value & Decision Trees.

From a single guess to a spread

Point estimates hide uncertainty. A Monte-Carlo simulation replaces a single guess with a distribution of possible values, then draws many random trials from it. Plotting the outcomes shows the full spread -- how wide the results can run -- and where the average lands. A common shape is the triangular distribution, defined by three points: an optimistic minimum, a most-likely mode, and a pessimistic maximum.

\[ \mu = \frac{min + mode + max}{3} \]

Reading the formula

The triangular distribution has a clean theoretical mean: add the three points and divide by three. For Northwind's build, min = $800,000, mode = $1,000,000, max = $1,500,000, so mean = (800,000 + 1,000,000 + 1,500,000) / 3 = 3,300,000 / 3 = $1,100,000. Run enough random trials and their average settles around $1,100,000, while the spread shows outcomes clustering near the mode with a longer tail toward the high end.

✨ Spread, not a promise

Monte-Carlo shows a range of outcomes around an average. It does not forecast a specific result or any real-world value. Here the inputs are illustrative teaching values; the lesson is the method -- how a distribution becomes a spread you can reason about.

🎮 Monte-Carlo Spread LIVE

Predict first: Predict first: min $800,000, mode $1,000,000, max $1,500,000 -- where does the mean sit?

Slide the optimistic, most-likely, and pessimistic values to watch a seeded triangular-sample histogram update, with the theoretical mean overlaid. Illustrative -- hypothetical, for learning only; not indicative of real outcomes.

📝 Worked example: A triangular cost estimate has min $800,000, mode $1,000,000, max $1,500,000. What is the theoretical mean?

1. mean = (min + mode + max) / 3.
2. = (800,000 + 1,000,000 + 1,500,000) / 3 = 3,300,000 / 3.
3. = $1,100,000.

✓ Mean = $1,100,000

✏️ Practice: A triangular estimate has min $600,000, most-likely $900,000, max $1,200,000. What is the mean?

💡 Hint

Add the three points and divide by three.

Solution

1. mean = (600,000 + 900,000 + 1,200,000) / 3.
2. = 2,700,000 / 3 = $900,000.

Check your understanding

1. Triangular estimate: min $800,000, mode $1,000,000, max $1,500,000. What is the mean?

mean = (800,000 + 1,000,000 + 1,500,000) / 3 = 3,300,000 / 3 = $1,100,000.

2. What does Monte-Carlo do?

Monte-Carlo draws many random trials from a distribution and shows the spread of outcomes plus where the average lands.

✅ Key takeaways

Monte-Carlo runs many random trials from a distribution to show the spread of outcomes.
For a triangular distribution the mean is (min + mode + max) / 3.
The spread shows how wide outcomes run around the average.

➡️ Monte-Carlo shows the full spread. Next we ask: which inputs move the outcome most? A tornado chart ranks them.

Ready for the next step? Back to the course outline →