Fun probability talk – Bertrand’s Paradox

This morning my son had an interesting question on a probability problem from an old AMC10:

Problem #13 from the 2011 AMC 10b

Here’s the problem:

Two real numbers are selected independently at random from the interval [-20, 10]. What is the probability that the product of those numbers is greater than zero?

Until today we’d mainly discussed probability in settings where there are a finite number of outcomes. In those cases I loosely define probability to be the number of “good” outcomes divided by the number of total outcomes. By “good” I mean the outcomes that you are looking for.

The difficulty in this problem is that there are an infinite number of outcomes, so you are dividing by infinity to calculate the probability if you use my loose definition. I’m really glad he asked me about this problem.


At the end of the last video I said that we’d now move on to Bertrand’s Paradox, but I put that idea aside for a second after I turned off the camera. There were still a couple of loose ends in my explanation of the prior problem and now that we’d be spending the whole morning on this problem, I thought it would be a good idea to at least mention some of these loose ends.

In particular, I wanted to show him one strange idea. We just showed that probability that the product is positive is 5/9. A similar argument will show that the probability that the product is negative is 4/9. The positive and negative probabilities add up to 1, but that doesn’t make sense, really, since there’s another case – the product can be zero!


Now we get to Bertrand’s Paradox. The question itself is surprisingly simple. Take a circle and inscribe an equilateral triangle in the circle. Now, randomly select a chord in the circle. What is the probability that this chord is longer than the side of the equilateral triangle?

Part 1: The probability is 1/4


Part 2: The probability is 1/2


Part 3: The probability is 1/3


So, geometric probability is strange 🙂 Really strange. Pretty much summed up by my son’s reaction around 1:50 in the last video – “This makes NO sense.” Indeed!

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