Bertrand's paradox

[wooded glade]

[sys3x]

Από τα καλλίτερα κανάλια στο yt αυτό, κόσμημα.

[Imperium]

Από τα καλλίτερα κανάλια στο yt αυτό, κόσμημα.

Πραγματι, κι απο πολλές απόψεις!
Έκανε διαθέσιμο και τον κόδικα που επινόησε για τα gfx.

[wooded glade]

It all depends what one means by random chord.
It depends what one means by randomness in general. We go to a street and we make a bet like this: Will the first person who turns the corner be bald or with hair ? After many trials a certain percentage will come up. But what if we go to another place, a place predominantly frequented by men ? Other percentage - bigger.
Could it be that 0.5 is the correct solution in the chord problem ?
Let's suppose that our circle is the circle x^2+y^2 = 1. That is the circle of radius 1 centred at the origin of a system of Cartesian coordinates.
Then we want a random straight line, this will be a line of the form Y = A.X + B.
A is the random tangent of an angle ranging from 0 to π/2 and B is also random but has to satisfy the condition B^2 < = 1 + A^2, so the line intersects the circle and a chord exists.
You can't get more random and unbiased than that.
Maybe I could proceed analyticaly from this point but I did n't. I used random number sampling and the result for the probability of length of chord >= sqr(3) is 0.5.
So that's the correct result and the other methods are biased.