# Joseph Bertrand Math Problem in Probability

#### Tyson

##### Member
An interesting problem in probability I came about today:

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?

Probably shouldn't look up the answer until you've worked it out on your own. And after you work it out, I'd suggest you think an extra moment.

#### mande

##### Member
I seem to be getting an answer of 1/3 fairly easily. But I will think an extra moment as you suggest.

EDIT: I wasn't able to see anything wrong with my argument, so I looked up wikipedia......pretty interesting.

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#### fanwuq

##### Member
Also got 1/3... but it could also be 1/2...?

Edit: Looked it up.
solution depends on method of selection

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#### LewisJ

##### Member
My first intuition is 1/3 - make one point of the chord a corner of the triangle; if the other point of the chord is between the two opposite corners of the triangle, it's longer, otherwise not, so you get 1/3.

But on second thought, what if the "random" chord is chosen differently? This depends a lot on how you randomly choose the chord. What if you just pick a random point in the circle and make that the midpoint of the chord? I'm not sure what the probability would be then but I'm confident it wouldn't be 1/4.

#### JBCM627

##### Member
Also got 1/3... but it could also be 1/2...?

Edit: Looked it up.
solution depends on method of selection
I got 1/2 initially. But yeah, I looked it up too... interesting.