The value of Pi (π)

1. Results of a single experiment:

Parameters:

D = Distance between the parallel lines.

L = Needle's lenght.

N= Number of needles (Attempts).

Variables:

y = Distance between the center of the needle and one of the lines.

$\theta$ = Angle that the needle makes with one of the lines.

Equation to estimate the value of π.

$\pi=\frac{2L}{fD}$

For this experiment:

Distance between lines (D) = {{outputs.distance}} cm

Needle lenght (L) = {{outputs.lenght_needle}} cm

Numbers of needles (N) = {{outputs.attempts}}

Outputs:

Pi: {{ outputs.PI}}

Probability to cross a line: {{ outputs.Firstprobability}}

Probability Graph

Picture a graph of y the needle center, plotted against θ, the needle orientation. As before, take the range of y to be from 0 to D and that of θ to be from 0 to 180° (π radians). Plot the lines given by y=L/2sin(θ)= and y=D-L/2sin(θ). Now we see curves and areas! We can see that the shaded areas represent values of y and θ for which we scored hits in our simulation. Expressed as a fraction of the rectangular area enclosed within the ranges 0 to D (y values) and 0 to π (θ values), they give us the value of f.

Representing the shaded areas by A, we have:

$f=\frac{A}{\pi D}$

In this case, the shaded area is given by: $A=2 \cdot \int^{\pi}_{0} (\frac{L}{2}) sin\theta d\theta$ , or $A=2L$ , so that $f$ becomes:

$f=\frac{2L}{\pi D}$

For our experiment:

{{ outputs.plot2}}

2. Precision, results for {{outputs.precision}} experiments:

The next figure shows the resulting scatter diagram of estimates obtained from {{outputs.precision}} trials. We can see that the estimates are scattered about the true value (blue dotted line).

{{outputs.plot}}

By generating a histogram of the resulting estimates, we can get a better sense of the inderlying variability.

For the {{outputs.precision}} trials, we have the following data:

Overall mean ( $\mu$ ) = {{outputs.mean}}

Standar Deviation ( $\sigma$ ) = {{outputs.standar}}

{{ outputs.plot1}}

Finding π (Pi) - Buffon’s Needle

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