In a circle of radius R , chords are drawn parallel to a…
In a circle of radius $R$, chords are drawn parallel to a given direction. What is the probability that the length of a randomly chosen chord is not greater than $r$, if all positions of the intersection point of the chord with the diameter perpendicular to the given direction are equally likely?
Explanation of Method
A chord is determined by the point where it crosses the perpendicular diameter; call its distance from the centre $x$, uniform on $[-R,R]$. By Pythagoras the chord has length $2\sqrt{R^2-x^2}$. The chord is short exactly when it is far from the centre, i.e. when $|x|$ exceeds a critical distance $c$. The probability is the fraction of the diameter where $|x|\ge c$.
Step 1: Length of a chord
Let the chord cross the perpendicular diameter at distance $x$ from the centre, $-R\le x\le R$. Half of the chord, the distance $x$ and the radius $R$ form a right triangle, so
\[ \text{length} = 2\sqrt{R^2-x^2}. \]
- crossing points giving a chord of length $\le r$
- some chords parallel to the given direction
Step 2: When is the chord short?
\[ 2\sqrt{R^2-x^2}\le r \iff R^2-x^2\le\frac{r^2}{4} \iff |x|\ge c, \qquad c=\sqrt{R^2-\frac{r^2}{4}}. \]
Step 3: The probability
$x$ is uniform on $[-R,R]$ (length $2R$). The favourable part $\{c\le|x|\le R\}$ has length $2(R-c)$:
\[ \boxed{P = \frac{2(R-c)}{2R} = 1-\sqrt{1-\frac{r^2}{4R^2}}, \qquad 0\le r\le 2R.} \]
Step 4: Examples
| $r$ | $P$ |
|---|---|
| $R$ (chord not longer than the radius) | $1-\tfrac{\sqrt3}2\approx0.134$ |
| $R\sqrt2$ (side of the inscribed square) | $1-\tfrac{1}{\sqrt2}\approx0.293$ |
| $R\sqrt3$ (side of the inscribed triangle) | $1-\tfrac12=0.5$ |
| $2R$ | $1$ (every chord is at most a diameter) |
Remark (Bertrand's paradox). The answer depends on what "randomly chosen chord" means. Here the crossing point with the diameter is uniform. If instead the midpoint of the chord were uniform in the disc, or the two endpoints uniform on the circle, the probabilities would be different. That is why the statement specifies the model.
Result
| Answer | |
|---|---|
| Chord length | $2\sqrt{R^2-x^2}$, $x$ uniform on $[-R,R]$ |
| Probability | $P(\text{length}\le r)=1-\sqrt{1-r^2/(4R^2)}$ |