Results
| Sample Size: | - |
| Points Inside: | - |
| π Estimate: | - |
| True π: | - |
| Absolute Error: | - |
| Standard Error: | - |
Estimating π
monte-carlo
probability
visualization
Use Monte Carlo methods to estimate the value of π by randomly sampling points in a square.
1,000
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About This App
This app demonstrates a simple application of Monte Carlo simulation: estimating π by random sampling. The method works by comparing areas:
- Consider a unit circle (radius = 1) inscribed in a square (side = 2)
- The circle has area \(\pi r^2 = \pi\)
- The square has area \((2r)^2 = 4\)
- The ratio of areas is \(\frac{\pi}{4}\)
By randomly generating points uniformly in the square and counting how many fall inside the circle, we can estimate this ratio:
\[\frac{\text{points inside circle}}{\text{total points}} \approx \frac{\pi}{4}\]
Therefore:
\[\pi \approx 4 \cdot \frac{\text{points inside circle}}{\text{total points}}\]
As we increase the number of random samples, the Law of Large Numbers guarantees that our estimate converges to the true value of π.