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bonanova

Question

Last night at Morty's, Alex asked his buddies:

What is the probability that a random chord drawn through a circle

has length greater than the side of an inscribed equilateral triangle?

After scribbling a moment on the classy new, M - monogrammed napkins, each of them had an answer:

Ian quickly announced, it's 1/4.

Jamie was more optimistic with his result of 1/3.

Davie smiled and said, If ya do it truly randomly, it'll happen 1/2 the time.

Which, if any, of them is right?

This is one puzzle where ambiguity in the OP is not only OK, it's necessary. :blink:

Give reasons, and please use spoilers.

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well if thats true then I cant say much except, whats my mistake? I cant find one in either of ours. I think mine is more reliable(who would have guessed) because I choose two random points then figure it out. Where you take a subset of chords and generalize. Oh and you need sqrt(3)*circleRadius not divide

but anyway I guess ill program this and see because I don't get where a mistake is in either

It didn't matter since i used r=1...but you're right. Brain fart...

Hehe... You really need to read the whole thread....neither of us have made a mistake. It matters how you define how to generate a random chord.

So far we have come up with:

a random r and random theta, and that is the midpoint of the chord. (random polar coordinates) = 1/2 << method i just gave

A random x,y (random cartesian coordinates) as the midpoint of the chord = 1/4

Two random polar coordinates (random thetas) where r=radius (random angle difference) around circumfrence = 1/3 << method you used

Oh, and Deegee's method, which is pretty darned close to 1, take a random x,y outside of the circle and the chord is defined by the two tangent intersections.

Edited by tpaxatb

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gotcha sorry i meant to go back and read the other posts but got sidetracked by another problem.. my bad

Thanx for the explanation

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Guest

How about this:

How do you draw a random line in a given XY plane. Do you "draw" all possible lines of all possible lengths and then choose a random line? Or do you choose two random points and connect them? here's an example: you draw a random triangle in XY plane. What is the probability that origin (0,0) is contained in the triangle? Well, well... nice idea for a new topic on the forum!

Without limits to your XY plane then there is as close to 0% chance as possible that the triangle will contain the coordinates (0,0). This is simply because there is an infinite amount of space in which this triangle can lie.

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