Monte Carlo simulation to estimate Pi

The Challenge!

Imagine if you will that you are playing a game of darts with your friend. The dartboard is hanging on a square piece of wood cut with the same diameter. Your friend boasts that they bet they could calculate the number pi by throwing darts randomly at the dartbard.

You are not sure you believe them, but decide to take them up on the challenge!

[1] 3.16

Lots of darts where thrown!

[1] 3.1524

Like 30,000 darts!

[1] 3.1384

You notice in the flurry of darts being thrown that your friend has been counting how many darts hit the inside of the circle, verses the outside. He keeps saying numbers that sound close to pi. You think you understand what the game is.

Lets look at the relationship of the unit circle and unit square. We start the familiar example of finding the area of a circle, which is pi*r^2. Similarly the area of the square is defined as 2*r^2.

The ratio of the area of the circle to the area of the square is:

area_cr = pi*r^2
area_sq = 2*r^2

# The ratio of areas
p = area_cr/area_sq 
  = (pi*r^2)/(2*r^2)
  = pi/4

[1] 0.7853982

If we multiply that constant by 4, we just get pi, 3.1415927.

As you can see from the previous slide the radius for a unit square and a circle cancel the term out always leaving the area inside to the outside of the circle a constant ratio pi/4 which is 0.7853982.

Another interesting to note, Accuracy of estimating pi goes up for large values of N. This is how our friend can caculate an estimate of pi. The more darts our friend throws, the closer we get to the actual value of pi.

The Shiny app you can throw darts yourself! https://zombieprocess.shinyapps.io/pi-monte-carlo