Picture a floor with parallel lines and a needle thrown onto it again and again, with no circle anywhere in sight. The share of throws that land across a line depends on the length of the needle and the spacing of the lines. What else is hiding in that ratio?
, by way of the angle. Two random quantities decide each throw: the distance of the needle’s center from the nearest line, and its angle. Every combination is a point in a rectangle, and the crossings fill a region bounded by a sine curve, so pi arrives through the integral. Which makes it a reversal: instead of using a known number to find a probability, you use a measured probability to find the number.
The setup of is almost suspiciously simple. A floor with parallel lines a fixed distance apart, and a needle of known length, thrown over and over. You count how often it lands across a line, and from the ratio you calculate pi.
The obvious question is where pi comes from at all, since there is no circle here. The answer is the angle: the needle lands in a random direction, and working out when it crosses a line means summing over all the angles. Any such sum brings pi with it.
More precisely it is , and two random quantities decide it: the distance of the needle’s center from the nearest line, and its angle. Every combination is represented by a point in a rectangle, and the cases where the needle crosses fill a region bounded by a sine curve, which is where the ratio comes from.
What is interesting here is not the calculation but the reversal: instead of using a known number to compute a probability, you use a measured probability to discover the number. That is the gateway to , which computes exact quantities from many random trials. And the price is visible here: the accuracy improves only as the square root of the number of throws, so a few throws will land a long way off.
wat. Throw a needle often enough and the geometry starts whispering pi.
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It is a fine way to see a mathematical constant appear out of random measurement, and an early gateway to the ideas behind Monte Carlo.
How do we know?
The claim follows from a calculation of geometric probability over the needle’s position and angle. Experiments and simulations illustrate it, and the formula itself is mathematical.
What is still unknown
A good estimate needs a great many throws and clean measurement. A few throws can give a result a long way from pi.
Learn something?
Where to next
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Can three dice each beat the next one around the circle?probability
What if the famous study disproving the hot hand had the maths wrong?probability