Probability Three Points On A Line & Measurement Paradoxes

Reader Neil Taylor points us to the video above, which is a jumping point for the larger question (be sure to watch the video first):

In [the video] Professor Stankova makes a statement along the lines that the probability of having three points form a line is zero.

I was fascinated by this statement — I presume it simply comes from a point being a genuinely insignificant part of a continuum.

What confuses me though is that given any two points we can construct a third which is guaranteed to lie on the line between them — via an Euler line for example.

The first two points form a triangle “centred” (via different definitions of centre) by these two points and then the third “centre” can be constructed which will by definition lie on the line through the first two points.

A certain way to reconstruct an event with zero probability.

What a wonderful paradox.

It’s not quite a paradox, because the Euler line is constructed via set rules. That the Euler line exists is proven (graphically and loosely in the video) via simpler arguments. Those three points, and all the other points on the line are there on purpose, if you like. And it is only one example of how to construct a line, which is defined as having infinitely many points which “lies equally with respect to the points on itself”, as Euclid said.

Now any two points can be said to lie on a line (mathematically—we’re not talking physics here). Take a third point and “drop it”, as suggested by the video. What is the probability that this interloper lies on the line?

Well, it’s a bad question. We don’t have enough information to discover. One of our premises is “drop it”. What exactly does “drop it” mean? We need to be rigorous in our definition since this is mathematics. Let’s start with a simpler problem and figure it out, and see how it fits in with this harder one.

Imagine a grid, size N x M, with both N and M finite, with slots which may or may not be equally sized, into which we can either place a wooden pin or nothing. Pick any two slots and place a pin in each. Needed next is the definition of a “line”, made by a path along the pins, on this grid. This line will be porous, and possibly jagged, because we don’t have infinitely many points. Use the same definition your computer screen uses in drawing lines with lighted pixels instead of pins (which are jagged, porous lines). Screen-lines only appear like mathematical-lines because N and M are large and your eyes small.

Anyway, with this definition in hand (whatever it is its premises will be about finite, discrete points) it’s now easy to calculate the probability of filling an unused slot that lies along the grid-line—as long as we have “drop it” defined. Perhaps the most natural definition is, “There will be X unused slots along the line as defined by the two present pins, and the third pin can be in any slot on the grid”. X of course is finite, too. The probability, conditional on all this evidence, is X/(NM – 2), where the subtraction is due to the initial two pins. (Some pedant out there will be situations where NM < 3.)

The next step is to increase N and M, not to infinity, but so that they are very large indeed. X will still be defined, because our grid-line and “drop it” are still defined. The probability is still X/(NM – 2). Finite can be very large indeed. N and M can be googols to the googol to the googol to the googol to the googol to etc. and still be finite. The probability will always be X/(NM – 2) no matter how large N and M grow, as long as they are finite (X depends on N and M and the definition of the grid-line).

Isn’t that interesting? Since N and M (and X) can soar to the heights, there doesn’t seem to be much need (or desire) for infinity. So there are two considerations. (1) What happens when N and M, and perforce X, go to infinity? (2) What if we confuse “drop it” with physics?

To infinity and beyond

Things do not just “go to” infinity, they go there via certain route and at a certain pace. Here, three things are heading off into the great beyond, and we need to be careful to specify with excruciating precision the route and pace of all three items, N, M, and X. We simply cannot rely on intuition because Infinity is too bizarre a place. It will bamboozle you every time. See Chapter 10 of Jaynes’s probability masterpiece for how infinity plays tricks on the finest minds.

One form of intuition suggests that our trio, N, M, and X, journey together at an even pace, in which case the probability X/(NM – 2) scoots toward zero, which is what the video suggested. But there are many other possibilities, and in some of these the probability changes.

Physics and measurement

Any system of measurement—and I go into this (and infinity) in more detail in Uncertainty: The Soul of Modeling, Probability & Statistics—is necessarily finite and discrete. Also remember probability is not physical, i.e. it is not ontic but epistemological, a matter of thought only. Probability can be used to speak about infinity, but measurement which confirms projects will only be finite and discrete.

Any real life physics problem should and must start with how we can measure it, thus we’re in the realm of finite and discrete, where calculations of probability involve only counting. If we think, based on other arguments, there is an infinite-continuous reality beneath the measurements (or a finer, and larger discrete one than our coarse measurements), then we have to suppose a path and a rate at which our measurements head towards this reality. Else, as we saw above, our probability calculations will be wrong.

Don’t be confused about “drop it”, or “drop randomly”, or “drop fairly” or some such other distraction. All that “random” and “fair” business leads to tautological arguments, or which reassert the probability above. On the other hand, if you’re imagining a physical pin bouncing around, you have to specify forces and equations of motion and so on, which if they don’t lead back to the original probability, will lead to something else easily countable—once the definition of these forces are in hand (and where “easily” is relative).

“Random”, “change”, “fair”, and all the rest are like Infinity. They are places where the intuition goes to die.


  1. Neil Taylor

    Prof Briggs, Thank you so much for answering my question. It is much appreciated.

  2. DAV

    The probability will always be X/(NM – 2) no matter how large N and M grow, as long as they are finite.

    Yes it can be confusing. You still need to specify that the cells will be selected without bias. Say by numbering them then picking numbers out of a jar without looking. I’m fairly sure this is what is meant by “drop it”.

    Interestingly, the cardinality of the set of points in a number line is the same as the cardinality of the set of points of a plane. So, when you go to infinity, X/(NM-2) becomes bizarre.

  3. JH

    Forget the nonsensical re-definition of a “line” and the question of whether a line can be said to be to be porous, let me give you a hint as to why your explanation is wrong. Regardless of the grid size, denoted by NxM in the post (be it square foot or square cm or whatever unit), assuming both N and M are positive, the probability of any three random points falling in a line is zero.

    And, if you have to alter the definition to answer a question, you are not answering the question.

    Forget whether the slots need to be of equal size, the number of slots in the grid may not be NM. It can be X+2, on this case, the probability is X/(NM-2) goes to 1 as X goes to infinity.

  4. Joy

    Wow what a coincidence. I’m behind reading this post today but
    Buzz lightyear, my old maths teacher and triangles; you had to be here.
    She emailed to say something important today and some things seem to fall into single points and in patterns.
    I also learned two new things today reading this and looking at the video and following it up. A light went on!
    Cryptic but so what?
    I now realise the part I missed in maths when I was younger and I must remember to tell her when I see her.
    Not that she cares much for maths these days. Memories are more precious.

  5. Fr. John Rickert, FSSP

    Perhaps I am missing something very basic to the intent of the original question, but when an event space is infinite, probability 0 does not mean impossible and probability 1 does not mean necessary. For example, if I say, “Guess the real number I am thinking of,” your probability is 0 unless I’ve given you at least some more information, but it is still possible that you could guess it.

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