The Idea Shop: A Barroom Method for Estimating Pi

A Barroom Method for Estimating Pi

Here's a method for estimating pi that basically requires a dartboard, faith in the relative frequency interpretation of probability, and plenty of time.

Step one, build yourself a dartboard that looks like this:

The area of the circular dart board is $\inline \pi r^2$ . And the area of the square behind the board is $\inline (2r)^2 = 4r^2$ .

Imagine throwing darts at this board. What's the probability it will land inside the circle? Easy enough -- it's the area of the circle divided by the area of the square, or

$P(dart \ in \ circle) = \frac{\pi r^2}{4r^2} = \frac{\pi}{4}$

$\Rightarrow \pi = 4P(dart \ in \ circle)$

Now we have an expression we can use to estimate pi. The process is simple.

Start throwing darts at the board. Be sure to throw randomly -- you'll need a uniform distribution of throws. Count the percentage that fall inside the circle. Then multiply this percentage by four. The result will be pretty close to pi. The more darts you throw, the better the estimate will get.

For those who want to see how this works in a spreadsheet, here's an Excel file with a simulation for n = 500 and n = 10,000 using random numbers from random.org for the x and y coordinates of the dart throws.

Update: A reader via email suggests another interesting method known as "Buffon's Needle Problem," which you can read about here.

Posted by Andrew on Monday May 11, 2009 | Feedback?

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