Buffon’s needle

The plane is ruled by parallel linesMathworldPlanetmath 2 inches apart and a 1-inch long needle is dropped at random on the plane. What is the probability that it hits parallel lines?
The first issue is to find some appropriate probability spaceMathworldPlanetmath (Ω,,P). For this,

  • h= distance from the center of the needle to the nearest line

  • θ= the angle that the needle makes with the horizontal ranging from 0 to π2.

These fully determine the position of the needle. Let us next take the

  1. 1.

    The probability space is Ω=[0,1]×[0,π2)

  2. 2.

    The probability of an event B is denoted by P[B] is equal to areaofBπ2

Now we denote by A the event that the needle hits a horizontal line. It is easily seen that this happens when sinθh1/2. Consequently A={(θ,h)Ω:hsinθ2} and then we get P[A]=2π0π212sinθdθ=1π

In general case, when the length of needle is l and the distance of parallel lines is d provided that l<d, the probability we want is 2lπd. This is obvious just taking the l/d-point from one edge instead of the center of the needle.

Title Buffon’s needle
Canonical name BuffonsNeedle
Date of creation 2013-03-22 16:09:28
Last modified on 2013-03-22 16:09:28
Owner georgiosl (7242)
Last modified by georgiosl (7242)
Numerical id 8
Author georgiosl (7242)
Entry type Definition
Classification msc 60D05
Classification msc 60-00