A plane is ruled by parallel lines at distance d from each…
A plane is ruled by parallel lines at distance $d$ from each other. Find the probability that a needle of length $l$ $(l\lt d)$ thrown at random crosses one of the lines (Buffon's problem).
Explanation of Method
The position of the needle is described by two numbers: the distance $x$ from its midpoint to the nearest line (uniform on $[0,\frac d2]$) and the angle $\varphi$ between the needle and the lines (uniform on $[0,\pi]$). The needle crosses the line when the half-needle reaches it: $x\le\frac l2\sin\varphi$. So the probability is the area under a sine curve divided by the area of the rectangle $[0,\pi]\times[0,\frac d2]$.
Step 1: Describing the position
- $x$ = distance from the midpoint of the needle to the nearest line, uniform on $[0,\frac d2]$;
- $\varphi$ = angle between the needle and the lines, uniform on $[0,\pi]$;
- $x$ and $\varphi$ are independent, so $(\varphi,x)$ is uniform in a rectangle of area $\pi\cdot\frac d2$.
Step 2: When does the needle cross a line?
The half-needle of length $\frac l2$ reaches a vertical distance $\frac l2\sin\varphi$ from the midpoint. It crosses the nearest line iff
\[ x\le\frac l2\sin\varphi. \]
(Because $l\lt d$, it can cross at most one line.)
- needle crosses a line: $x\le\tfrac l2\sin\varphi$
Step 3: The area under the curve
\[ \int_0^\pi\frac l2\sin\varphi\,d\varphi = \frac l2\big[-\cos\varphi\big]_0^\pi = l. \]
Step 4: The probability
\[ \boxed{P = \frac{l}{\pi\cdot\frac d2} = \frac{2l}{\pi d}.} \]
Step 5: Estimating $\pi$
The formula contains $\pi$, so throwing a needle many times gives an estimate: if it crosses $k$ times in $N$ throws, then $\pi\approx\dfrac{2lN}{dk}$. For $l=d$ (a limiting case) $P=\frac2\pi\approx0.637$.
See Problem 77 for the generalisation to any convex contour: $P = \frac{\text{perimeter}}{\pi\cdot\text{distance}}$; a needle is a contour of perimeter $2l$.
Result
| Answer | |
|---|---|
| Crossing condition | $x\le\frac l2\sin\varphi$ |
| Probability | $P=\dfrac{2l}{\pi d}$ |