Compare random-chord probabilities under uniform endpoints, uniform center distance and disk-area midpoint sampling.
How to use this tool
- Choose the event or model and review the assumptions below.
- Enter the labeled inputs.
- Select Calculate and read the labeled outputs.
- Change an input and calculate again; Reset restores the example inputs.
Calculation method
For threshold cR, probabilities are 1 − 2 asin(c/2)/π (endpoints), √(1−c²/4) (uniform distance), and 1−c²/4 (uniform disk midpoint).
Assumptions and limits
The sampling mechanism defines the probability. Threshold c is between 0 and 2. At the inscribed equilateral-triangle side, results are 1/3, 1/2 and 1/4 respectively.
Worked example
Set the threshold-to-radius ratio to √3, approximately 1.7320508075688772. This is the side length of an equilateral triangle inscribed in a unit-radius circle. The probability of a longer chord is 1/3 for uniform boundary endpoints, 1/2 for uniform perpendicular distance, and 1/4 for a midpoint uniform over disk area. Different answers reflect different distributions on chords, not inconsistent arithmetic.
Worked reading and practical questions
Browse Statistics calculators or explore other tools on the GreenCalc homepage.
Frequently asked questions
What does this calculator assume?
The sampling mechanism defines the probability. Threshold c is between 0 and 2. At the inscribed equilateral-triangle side, results are 1/3, 1/2 and 1/4 respectively.
How do I enter and update the inputs?
Use the labeled fields, then select Calculate. Percent fields use 0 to 100. After editing, calculate again; Reset restores the example inputs.