Consider a simple case in which the lines are 1 cm apart and the needle is 1 cm in length. After many drops the probability of the needle lying across a line is found to be very close to 2/π. Why? There are two variables: the distance from the center of the needle to the closest line, d, which can vary between 0 and 0.5 cm, and the angle, θ, at which the needle falls with respect to the lines, which can be vary between 0 to 180°. The needle will hit a line if d ½ sin θ. In a plot of d against ½ sin θ, the values on or below the curve represent a hit; thus, the probability of a success is the ratio of the area below the curve to the area of entire rectangle. The area below the curve is given by the integral of ½ sin θ from 0 to π, which is π. The area of the rectangle is π. So, the probability of a hit is 1/(π/2) or 2/π (about 0.637). Dropping a needle many times on to lined paper gives an interesting (but slow) way to find π.
This kind of probabilistic means of performing calculations is the basis of a technique known as the Monte Carlo method.
Related category PROBABILITY AND STATISTICS
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