## Monday, June 20, 2011

### Ngon Dartboard Probability Simulation

While I was working on the extension problem, Andy Rundquist sent me the results of his simulations along with the screencast showing how he did it in Mathematica.

Evaluating the formula from the last post
\begin{align*} P \left ( d_{c}\lt d_{e} \right) &= \frac{ tan^3 \frac{\pi}{2n} + 3 tan \frac{\pi}{2n} }{ 6 tan \frac{\pi}{n} } \\ \end{align*}
... for several different n, we have:
n Probability Andy's Simulation
3 .1851851852

4 .2189514165

5 .2314757303

6 .237604307

7 .2410913279

8 .2432751885
9 .2447375774
10 .2457666208
Most of the results are fairly close. I think seeing the results of Andy's simulations definitely gave me a little more confidence about posting my attempted solution online.

Some of the numbers from the simulations are a little off and that's expected. Run the simulation a few more times and you'll get a slightly different result every time. They fluctuate around the "true" proportion of darts that will land closer to the center. For the simulation diagrams above, Andy used 30,000 as the number of trials.

Extension (Statistics): What is the number of trials that should be used for simulations if we want most of our simulations to be near our "true" proportion (e.g. within 0.01, 0.001, 0.0001, etc)?

Square Dartboard Probability
Ngon Dartboard Probability
Ngon Dartboard Probability Simulation
Ngon Dartboard Probability Limit