# What is Bertrand's paradox

What is the probability that a randomly selected chord of the unit circle is longer than one side of an equilateral triangle inscribed in this circle?

### version 1

Each chord can be clearly identified by its center point (the chord is then formed normal to the line connecting the circle center point). The center point P of the chord is given by the Cartesian coordinates (x, y). The event space Ω

_{1}is given in this way by.**Event A**: The selected chord is larger than the side of the triangle;_{1}**The probability for A**_{1}amounts**.****task**Move the**Point p**and the**upper point of the triangle**to familiarize yourself with the problem. Display the simulation.### Variant 2

The center of the chord can also be given by its polar coordinates (r; φ). The event space Ω

_{2}is given in this way by.**Event A**: The selected chord is larger than the side of the triangle;_{2}**The probability for A**_{2}amounts**.****task**Move the**Point p**and the top tip of the triangle. Display the simulation.### Variation 3

The chord can also be given by a point on the edge of the circle (i.e. by the arc length b) and an angle α to the tangent. The event space Ω

_{3}is given in this way by.**Event A**: The selected chord is larger than the side of the triangle;_{3}**The probability for A**_{3}amounts**.****task**Move the**Point p**and the**upper point of the triangle**as well as the slider for the angle α. Display the simulation.The

**Paradox**is that there are different probabilities depending on the choice of tendon. The reason for this lies in the fact that the random selection of a tendon is not precisely specified.- What are the most ingenious inventions in history
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