The needle in the picture misses the line. The needle will hit the line if the closest distance to a line (D) is less than or equal to 1/2 times the sine of theta. That is, D ≤ (1/2)sin(θ). How often will this occur?
In the graph below, we plot D along the ordinate and (1/2)sin(θ) along the abscissa. The values on or below the curve represent a hit (D ≤ (1/2)sin(θ)). Thus, the probability of a success it the ratio shaded area to the entire rectangle. What is this to value?
To calculate pi from the needle drops, simply take the number of drops and multiply it by two, then divide by the number of hits, or 2(total drops)/(number of hits) = π (approximately).
- Madotto, Federico. Buffon's (short) Needle Problem. Digital image. Wikipedia. Wikimedia Foundation, 21 June 2012. Web. 17 Feb. 2015. <http://en.wikipedia.org/wiki/File:Buffon%27s_(short)_needle_problem.gif>.
- Reese, George. "Buffon's Needle." Mathematics, Science, and Technology Education. University of Illinois, n.d. Web. 17 Feb. 2015.
- Wikipedia Contributors. "Buffon's Needle." Wikipedia. Wikimedia Foundation, 14 Apr. 2014. Web. 17 Feb. 2015.