Another bit of trigonometry: two different methods of random chord probabilities.

Algebra Level pending

The problem question: What is the Inside answer minus the Outside answer? If they are the same, then, enter 0.

Two different methods of computing probability of a random chord being $\geq 1$ in a radius 1 circle on an Euclidean plane: with origin at {0,0}:

Outside: Fraction of circumference such that the Euclidean distance from {1,0} is $\geq 1$ .
Inside: Fraction of the {0,0} to {1,0} radius such that a chord perpendicular to the radius is $\geq 1$ in total length.

The difference magnitude is less than 1. Six decimal places to the right of the decimal point were entered as the answer.

I solved this problem by integration and the assumption of uniform distribution along the circumference (Outside) and radius (Inside).

The answer is 0.199358.

1 solution

A Former Brilliant Member
Apr 28, 2019

Outside: Solve $\text{EuclideanDistance}[\{1,0\},\{\cos (\theta ),\sin (\theta )\}]=1\Rightarrow \theta =\frac{\pi }{3}\lor \theta =\frac{5 \pi }{3}$ . $\frac{\frac{5 \pi }{3}-\frac{\pi }{3}}{2 \pi } \Rightarrow \frac23$
Inside: $\cos \left(\sin ^{-1}\left(\frac{1}{2}\right)\right)\Rightarrow \frac{\sqrt{3}}{2}$ .
$\frac{\sqrt{3}}{2}-\frac{2}{3}\Rightarrow 0.199358737117772$