Chords chords chords

Probability Level 4

In the circumference of a circle of radius $r$ , two points are chosen with a uniform distribution. The probability that the length of the chord between these two points is less than $r$ is $\frac{ a}{b}$ for coprime positive integers $a, b$ .

Find $a+3b$ .

This problem was from CB Paul Science quiz (Stage round) where it is expected to be solved in 40 seconds.

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The answer is 10.

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Julian Poon
Jan 30, 2015

Here's a solution without words:

Latex!!!

For those who didn't understand, here's another clue: $P=\frac{\theta }{90°} = \frac{1}{3} = \frac{a}{b}$

Hm, I think drawing a regular hexagon would make much more sense as a proof without words.

Note that this is technically a 1-d geom probability, because you are chasing after the length of the circumference, instead of the area of the circle.

Calvin Lin Staff - 6 years, 4 months ago

Oh. Thanks for clarifying!

Julian Poon - 6 years, 4 months ago

The question as presently worded, though, can be interpreted in several ways, as Michael points out in his post. The use of the word "randomly" will need some clarification.

Brian Charlesworth - 6 years, 4 months ago

Michael Mendrin
Jan 30, 2015

Poon, I hate to bust a good party, but I have very strong reservations about this problem. What does "a chord drawn randomly on a circle" of radius r mean? One can get different answers to this one depend on the definition of "a chord drawn randomly on a circle". There are quite a few geometrical probability problems that suffer from lack of precision about how certain elements are "chosen randomly", such as the classic "Bertrand Paradox", which is similar to this problem here.

You need to be more specific about how the chord is "chosen randomly". You can say, "Given a point chosen randomly on the circumference on the circle, draw a line at a random angle through this point producing a chord".

I had to try a couple of answers depending on how the chord is selected before I luckily hit on the "correct" answer.

Would "Randomly choose two points on the circumference of the circle and draw a chord between them. What is the probability that the length of this chord is less than $r$ ?" be an appropriate rephrasing? It was the only interpretation I tried that yielded a rational solution.

Brian Charlesworth - 6 years, 4 months ago

Brian, that's probably a nicer way to define the random chord for this problem. To first pick at random on the circumference and then draw a line at a random angle actually has the same distribution as picking two random points on the circumference and drawing a line through it. So, because you probably used your idea of a random chord, you came up with the correct answer for this problem as posted.

Michael Mendrin - 6 years, 4 months ago

Ah, right, they do have the same distribution. In any event, JP will have to clarify the wording.

Brian Charlesworth - 6 years, 4 months ago

Precisely! A chord can also be uniquely characterized by the distance of it's midpoint from the center. This in turn can be done in two ways: Pick the midpoint from among all interior points or just pick it's distance from the center, from 0 to $r$ .

Shashwat Shukla - 6 years, 4 months ago

Sorry for the ambiguity. Somebody helped me to edit the problem so this is no longer an issue. I'll take note of this in my later problems.

Julian Poon - 6 years, 4 months ago

Prakriti Bansal
Feb 20, 2015

Chords chords chords

This problem was from CB Paul Science quiz (Stage round) where it is expected to be solved in 40 seconds.

Try my Other Problems

The answer is 10.

3 solutions

Discussions for this problem are now closed

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