keeps throwing, faster and faster, up to 1,000,000 darts
Darts thrown0
Inside the quarter circle0
Outside0
Estimate of π–
Real π3.141593
Off by–
Correct decimals–
π ≈ 4 × inside ÷ thrown
The estimate as the darts add up
The green band is the typical error for that many darts, about 1.64 ÷ √darts. Your line wanders inside it, and the band narrows only slowly.
What one more correct digit costs
Darts
Typical error
Your error
The typical error shrinks with the square root of the number of darts: 100 times more darts make it 10 times smaller, which is one more correct decimal. That is why this method is easy, but slow: 1,000,000 darts pin π down only to about ±0.0016, which is two or three correct decimals.