Solving Problems Step by Step

In mathematics, as in everything, intuition is a double edged sword

Consider a disc of radius R comprising of infinitely many points distributed uniformly over its surface. What’s the average distance of a point from the centre of the disc?

A few days ago I described this probability puzzle which seemed to yield two different results depending upon which “intuition” I followed. Clearly, for an unambiguous math puzzle, there can be only one correct result. It turns out that there is a correct result, but the two solution approaches that I mention in that article are both wrong.

That one of the solutions yields the correct result is inadvertently a case of two wrongs making it right.

Recall that there is a geometric solution that yields the result 2R/3, which in fact is the correct answer. Here, using probability theory, I will only provide (what I believe is) the correct solution. I hope that it will be automatically clear why both the solutions mentioned in the older article are incorrect, in spite of the fact that both appeal equally strongly to intuition.

The problem states that the points are distributed uniformly, and over the surface of the disc. To cover the surface of the disc, we need two variables r and 𝜃. The probability density function (PDF) is therefore a function of these two variables. In concrete terms, the PDF is given by the following expression.

Moreover, we can only refer to the probability Pr(p ∈ X) denoting that a point p chosen at random comes from a specific area X on the disc.

Here dA is the differential of the area of the region. Since the above expression must evaluate to 1 over the entire disc, we have that c = 1/𝜋R². If we use a random variable Y to model the distance r of a point from the centre of the disc, the result can be obtained by computing the expected value of Y over the entire disc D.

This is the correct answer using the correct solution approach. And it wasn't that difficult. If only I had delayed my intuition and solved it like they taught us in school — step by step.