What Is Monte Carlo Probability?
Monte Carlo simulation estimates a probability by running a random experiment many times and counting how often the event happens. It is a practical method when the exact calculation is hard.
Choose dice, coin flips or the birthday problem, set the number of trials and a seed. The calculator runs the simulation, shows the estimate with a 95 percent interval, and prints the exact probability next to it.
A graph shows the running estimate settling toward the exact value as the number of trials grows.
How to Use the Monte Carlo Probability Calculator
- Pick the experiment and its size.
- Set the number of trials and a seed.
- Compare the estimate, its interval and the exact value.
Monte Carlo Probability Formula
Estimate
Standard error
Monte Carlo Probability Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| N | Number of trials | |
| p̂ | Estimated probability | hits / N |
| SE | Standard error of the estimate | √(p̂(1 − p̂)/N) |
| Seed | Starting value of the random generator | Same seed, same result |
Monte Carlo Probability Worked Example
With the example values loaded in the calculator above, the steps are:
- Event: Sum of 3 dice ≥ 12.
- Run 100,000 random trials with a seeded generator (seed 42), so the same inputs always give the same result.
- Count the trials where the event happened: 37420. Estimate = 37420 ÷ 100000 = 0.3742.
- Standard error = √(p̂(1 − p̂) ÷ N) = 0.00153028, so the 95% interval is 0.3712 to 0.3772.
- The exact value is 0.375, a difference of -0.0008.
Result: Estimated probability = 0.3742; Exact probability = 0.375.
Worked example: three dice, sum at least 12
Of the 216 equally likely rolls of three dice, 81 give a sum of 12 or more, so the exact probability is 81 ÷ 216 = 0.375.
A simulation with a hundred thousand trials should land within about 0.003 of that.
The birthday problem
With 23 people the chance that two share a birthday is just over 50 percent, assuming 365 equally likely days. Try it with 23 people to see the estimate settle near 0.507.
Where Monte Carlo Probability Is Used
Learning simulation
See how an estimate converges to the exact value.
Checking a probability
Verify a hand calculation for dice or coins.
Teaching the birthday problem
Show how quickly shared birthdays become likely.
Common Monte Carlo Probability Mistakes to Avoid
- More trials mean a more precise estimate, with diminishing returns.
- A fixed seed makes your result reproducible.
Monte Carlo Probability Terms and Definitions
- Monte Carlo method
- Estimating a quantity by repeated random sampling.
- Seed
- The starting value of the random number generator.
Monte Carlo Probability FAQs
What is a Monte Carlo simulation?
A way to estimate a probability or value by running a random experiment many times.
How many trials do I need?
It depends on the precision you want. The error falls like 1 over the square root of the number of trials.
Why does the estimate differ from the exact value?
Random variation. The difference is usually inside the 95% interval.
Why use a seed?
It makes the simulation repeatable. Change it to see how results vary.