Calculator Inputs
Example Data Table
| Goal | Mode | Main Inputs | Expected Output |
|---|---|---|---|
| Estimate pi | Estimate Pi | 10,000 trials, 95% confidence | Pi estimate with interval |
| Find area under curve | Estimate Definite Integral | f(x) = x², interval 0 to 1 | Approximate integral value |
| Measure risk chance | Estimate Probability | Normal mean 50, deviation 10, value ≥ 70 | Probability and percentiles |
| Model uncertain value | Estimate Probability | Triangular min 10, mode 25, max 60 | Mean, deviation, and chance |
Formula Used
Pi Simulation Formula
Pi is estimated with this formula:
π ≈ 4 × inside points / total points
Integral Simulation Formula
The calculator samples random x values over an interval.
Integral ≈ (b - a) × average f(x)
Probability Formula
The calculator counts successful simulated outcomes.
Probability ≈ successful trials / total trials
Confidence Interval Formula
The interval uses the standard error and z score.
Estimate ± z × standard error
How to Use This Calculator
Select a simulation type first. Use pi mode for geometric estimation. Use integral mode for curve area estimation. Use probability mode for uncertain value modeling. Enter the number of trials. More trials usually improve stability. Choose a confidence level. Add a random seed when you want repeatable results. Leave it blank for fresh random trials. Press the calculate button. The result appears above the form. Use CSV or PDF buttons to save the report.
Monte Carlo Simulation Guide
What This Calculator Does
Monte Carlo simulation uses random sampling to estimate an answer. It is useful when direct calculation is hard. The method repeats a simple experiment many times. Each run creates a possible result. The calculator then summarizes all simulated outcomes. This makes uncertainty easier to understand.
Why Random Trials Help
Many math problems involve unknown values. Some values follow a range. Some follow a bell curve. Others have a most likely value. Random trials let you test those cases quickly. They show the average result, spread, and chance of meeting a condition.
Common Uses
You can estimate pi with random points. You can estimate an integral by sampling a function. You can also estimate risk, success chance, or threshold probability. This is helpful in finance, engineering, statistics, planning, and classroom work.
Understanding Results
The estimated value is the main answer. The standard error shows sampling noise. A lower standard error means a more stable estimate. The confidence interval gives a likely range for the true value. A wider interval means more uncertainty.
Choosing Trial Count
Small trial counts run quickly. They may jump between results. Larger trial counts take more time. They usually produce smoother estimates. Start with ten thousand trials. Increase the value when accuracy matters. Use the same seed to compare settings fairly.
Distribution Options
Uniform distribution treats every value in a range equally. Normal distribution works for values around a mean. Triangular distribution is helpful when you know the low, high, and most likely value. Choose the option that best matches your situation.
Best Practice
Monte Carlo simulation does not replace exact formulas. It supports decisions when exact formulas are difficult. Always review inputs before trusting output. Use realistic ranges. Compare several runs. Export the report when you need to document assumptions, inputs, and results.
FAQs
What is a Monte Carlo simulation?
It is a method that uses random trials to estimate a value, probability, or range. It is useful when a problem has uncertainty or is difficult to solve exactly.
How many trials should I use?
Use at least 10,000 trials for general estimates. Use more trials when you need smoother results. Very high trial counts may take longer to process.
Why do results change after each run?
The calculator uses random samples. New samples can create slightly different results. Add a random seed when you want the same result repeated.
What does the confidence interval mean?
It shows a likely range around the estimate. A narrow interval means the estimate is more stable. A wide interval means more sampling uncertainty remains.
Which distribution should I choose?
Use uniform for equal chances across a range. Use normal for values around a mean. Use triangular when you know minimum, maximum, and most likely values.
Can this calculator estimate pi?
Yes. It places random points in a square and counts points inside a quarter circle. The ratio is then used to estimate pi.
Can I estimate an integral?
Yes. Choose the integral mode, select a function, and enter bounds. The calculator samples random x values and estimates the area under the curve.
Can I save my results?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a simple report that includes the main simulation results.