Why do not you get an estimation for the value of π?
The only thing you should do is to draw parallel straight lines on the sheet of paper with the distance of the length of the needle you have. Then you have to drop the needle on the sheet. With some simplification let’s separate the outcome depending on the case when the dropped needle crosses or does not touch any line you just drew. If you repeat this experiment by a couple of thousand times then you have to calculate the ratio of outcomes with crossing a line against the total number of trials. This will your estimation for π.
This approach was originally proposed by Georges-Louis Leclerc, Comte de Buffon in the 18th century. This is probably the most famous demonstration of application of geometry in managing of probability theory (or statistics).
Anyway, you have to spend some with this to get close to the actual accuracy to the current existing estimation (which is calculated today up to 12.1 trillion digits by Alexander J. Yee and Shigeru Kondo – according to their update on Number of digits of pi). (If you do not use trillions every day: the actual value of π contains 10^12 digits!!!)
As it is described by Wikipedia that this experimental design is absolutely suitable to commit the so called confirmation bias, a

“a tendency to search for, interpret, favour and recall information in a way that confirms one’s pre-existing beliefs or hypotheses, while giving disproportionately less consideration to alternative possibilities.”

Lazzarini, an Italian mathematician performed Buffon’s needle experiment with tossing a needle 3408 times and achieving the already “famous” approximation of 355/113 for π. He artificially set-up such an environment where he could expect 113*n/213 as the estimation (n denotes the length of the trial, that is the number of needle drops). He had to repeat the whole experiment only 16 times to reproduce the magic 355/113. However, it is important to note, that Lazzarini did not do anything wrong or unethically. He committed a bias – he was imprudent in some measure – but to avoid confirmation bias is a real challenge for majority of the researchers.
If you are interested in this experiment, you can check this with the help of easy-to-access tools, like Java applets or Flash. Probably the nicest solution can be found at http://www.ventrella.com/buffon/ (by Jeffrey Ventrella), while a simpler, but still very impressive solution can be found at http://www.metablake.com/pi.swf .