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### Sum of Two Arcs

```Date: 01/30/2003 at 20:46:53
From: Abishek
Subject: Probability

Hi Dr. Math,

This is the question:

Three points are taken at random on the circumference of a circle.
What is the chance that the sum of any two arcs so determined is
greater than the third?
```

```
Date: 01/31/2003 at 01:56:00
From: Doctor Greenie
Subject: Re: Probability

Hi, Abishek -

I enjoyed the mental exercise I got thinking this one through!

If the sum of two of the three arcs is greater than the third, then
the largest arc is less than half the circle - i.e., less than 180
degrees. So to solve your problem, let's look at the conditions under
which the largest of the three arcs is GREATER than 180 degrees.

Suppose we call the points A, B, and C, and suppose points A and B
have been chosen, and let x be the degree measure of minor arc AB
(i.e., of the smaller of the two arcs determined by points A and B).

Now consider diameters AD and BE (draw a picture...).  If the third
point C is on the same side of diameter AD as point B, then the
largest of the three arcs will be greater than 180 degrees. And if the
third point C is on the same side of diameter BE as point A, then the
largest of the three arcs will be greater than 180 degrees. So the
range where point C can be chosen so that the largest of the three
arcs is LESS than 180 degrees is the minor arc DE, whose measure is
the same as the measure of minor arc AB, which is x degrees.

So if the measure of minor arc AB is x degrees, then the probability P
(x) that the largest of the three arcs is less than 180 degrees is

P(x) = (x/360)

If we graph this probability function over its domain from 0 to 180
degrees, we get a straight line with P(0) = 0 and P(180) = 1/2.  The
overall probability that the largest of the three arcs is less than
180 degrees is the average value of the function P(x) over its
domain, which is 1/4.

So...

Given three points randomly chosen on a circle, the probability that
the largest of the three arcs is less than 180 degrees - that is, the
probability that the sum of any two arcs is greater than the third -
is 1/4.

I looked in the Dr. Math archives after I wrote this response,
performing a search using the keywords "probability point
circumference arc".  That search turned up this link which discusses
this same problem but never - as far as I could tell - comes to a

A Triangle in a Circle
http://mathforum.org/library/drmath/view/51787.html

- Doctor Greenie, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 02/01/2003 at 19:54:24
From: Abishek
Subject: Probability

I could follow till the part where you said p(x)= x/360, but after
that I don't understand why we have to take the average value for the
required probability.
```

```
Date: 02/02/2003 at 00:22:27
From: Doctor Greenie
Subject: Re: Probability

Hello, Abishek -

The statement

P(x) = x/360

says that the probability is x/360 that the largest of the three arcs
is less than 180 degrees if the minor arc between the first two of
the three points measures x degrees.

So, for example, if the minor arc between the first two points is 45
degrees, then the probability of choosing the third point so that the
largest arc is less than 180 degrees is 45/360 = 1/8.  Or, if the
minor arc between the first two points is 120 degrees, then the
probability of choosing the third point so that the largest arc is
less than 180 degrees is 120/360 = 1/3.

To find the overall probability that three points randomly chosen
will result in the largest of the three arcs measuring less than 180
degrees, we need to consider the probabilities P(x) over the whole
range of possible value of x, which is from 0 to 180 degrees.  The
probability function in this case is a linear function, with value 0
at x=0 and value 180/360 = 1/2 at x=180.  The average value of a
linear function over its range is the average of its values at the
two endpoints of its range -- in this case, the average of 0 and 1/2,
which is 1/4.

I hope this helps.  Please write back if you still have questions

- Doctor Greenie, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Euclidean Geometry
College Probability
High School Euclidean/Plane Geometry
High School Probability

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