On a circle of unit radius, pick a chord “uniformly at random.” What’s the probability its length is greater than square root of three?

let's make it a multiple choice question. your options are 1/2, 1/3, and 1/4. pause the video and see if you can solve this problem.

The surprising fact is that all three answers are correct. This is a paradox discovered by the French mathematician, Joseph Bertrand. I’ll explain the details later. First, let's compute these three different values.

To see why one-half is a correct answer, imagine a unit circle centered at O. Draw a chord AB of length square root of three, and then draw a line OD passing through the midpoint C of the chord AB. Since AB equals root three, AC equals root three over two. From basic geometry, we know the line OC is also perpendicular to the chord AB. Applying the Pythagoras theorem to triangle OAC, we find that OC equals one-half.

Here’s a result. if the perpendicular distance of a chord from the centre of a unit circle is greater than one half, then the chord’s length will be less than root three. Otherwise, the it will be greater than root three. So, the probability that a random chord is longer than root three is simply the ratio of length OC over length OD, which comes out to one-half.

Now let’s understand why one-third is also a correct answer. Once again, take a unit circle with a chord AB of length square root of three. From the center O, draw lines to the endpoints A and B. Using trigonometry, we can calculate the smaller of the two central angle AOB. Since OC equals one-half and OA equals one, the angle comes out to two times the inverse cosine of one-half, which is one hundred and twenty degrees.

Here's a result. If a chord on a unit circle makes a central angle less than one hundred and twenty degrees, its length will be less than root three. If the angle is greater than one hundred and twenty degrees, the chord’s length will exceed root three.
Therefore, the probability that a random chord is longer than root three is the fraction of angles greater than one hundred and twenty degrees out of the total one hundred and eighty degrees. That ratio is sixty over one eighty, which equals one-third.

Finally, let’s see why one-quarter is also a correct answer. Start with a unit circle and a chord AB of length square root of three. Draw a line from O through C; we already know that OC equals one-half. Now, draw a smaller circle centered at O with radius one-half.
Here’s a result. a chord on a unit circle will be shorter than root three if its midpoint lies outside this smaller circle. Otherwise, the chord will be longer than root three.
That means the probability is simply the ratio of the two areas — the area of the small circle to the area of the big circle. The small circle has area pi times one-half squared, and the big circle has area pi one squared. The pi cancels out, leaving one-quarter as the solution.

So why do we end up with so many answers to the same problem? As mentioned earlier, this is known as Bertrand paradox. in classical probability theory, When we deal with simple cases like rolling a die or tossing a coin, we rely on intuition and prior experiences to define equally likely events.

But with random chord on a circle, the situation is different. The sample space of all possible chords can be described in several natural ways: by their perpendicular distance from the center, by the central angle they subtend, or by the position of their midpoint. Each way gives a different meaning to the word “uniform” and yields different probability. the axiomatic probability framework fixes this by requiring specification of a measure which precisely defines how probability is distributed over the sample space. thereby, removing dependence on intuition and prior experiences.