Buffon's Needle

Question

A floor is ruled with parallel lines exactly one needle-length apart. You drop the needle so its center lands uniformly at random and its angle is uniform. What is the probability the needle crosses a line?

Accepted Answer

Normalize the spacing and needle length to 1. By symmetry, reduce to x sim Uleft[0, frac12right] (distance from center to the nearest line) and theta sim U[0, pi] (angle to the lines), independent. The needle's half-extent perpendicular to the lines is frac12sintheta, so it crosses iff x le tfrac12 sintheta. Conditioning on the angle: P(textcross mid theta) = fracfrac12sinthetafrac12 = sintheta. Averaging, P = frac1piint_0^pi sintheta , dtheta = frac2pi approx 0.6366. Approach 2, the noodle argument (no integral) Let f(ell) be the expected number of crossings for a rigid curve of length ell thrown at the lined floor. Chopping curves into pieces and adding, f is additive, hence linear: f(ell) = cell, for any shape. Calibrate with a circle of diameter 1 (circumference pi): it crosses the unit-spaced lines exactly twice, always. So cpi = 2, giving c = frac2pi. A straight needle of length 1 crosses 0 or 1 times, so its crossing probability equals its expected crossings: frac2pi. The noodle argument is the one to tell an interviewer: linearity of expectation applied to geometry, calibrated by a clever special case, replacing calculus entirely. The famous corollary Dropping n needles and

Buffon's Needle

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