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Strike a Chord (..Any Chord) (Posted on 2003-10-09) Difficulty: 4 of 5
What is the probability that a randomly drawn chord will be longer than the radius of the circle?

Prove it.

No Solution Yet Submitted by DJ    
Rating: 4.5263 (19 votes)

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Herd instinct. Comment 51 of 51 |

'What is the probability that a randomly drawn chord will be longer than the radius of the circle?' (DJ)

'Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random. What is the probability that the chord is longer than a side of the triangle?'(Bertrand)

A side of Bertrand's triangle is a chord with an angle of 120 degrees at O. A side of DJ's triangle is a chord with an angle of 60 degrees at O. So let's start from there:

1. Select a point A on the circle with centre O, radius 1
2. Construct a chord BC, length 1, perpendicular to that point. This is obviously one side of equilateral triangle BCO.
3. The height of BCO is then 3^(1/2)/2. We draw a new circle, radius 3^(1/2)/2, on O.
4. All the chords that are less than the radius of the circle will occur in the area between the old circle and the new one.
5. As a proportion of the whole area this is: 1-(3^(1/2)/2)^2, or 25%. So 75% of chords will be greater.
6. I accept that is not the only way to think about it. The arc BC is 1/6 of the circle. So if the second point of the chord is completely random, then 1/6 of the second points will be less than r, and 5/6 will be greater (Note: Not 1/3 - the second point can be clockwise or anticlockwise, but not both!) This method would give a probability of (250/3)%. But that is the somewhat different question: given point 1 on a circle, what is the probability of point 2 on the same circle being closer than the radius of the circle? We should not necessarily expect the answer to be the same.

Last but not least, a method similar to (1-5) is to simply ignore the area approach and take the result as a linear proportion of the total number of lines parallel to a diameter (since every chord is by definition parallel to some diameter). The result then is as 3^(1/2)/2 is to 1. this gives a result of 86.6025%. But again the same issue arises: what is the probability, given a diameter, of a chord perpendicular to it being longer than the radius? We should not expect the same answer.

So I conclude that 75% is the best answer, given the question as set.

Edited on May 29, 2016, 3:22 am
  Posted by broll on 2016-05-29 01:14:59

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