Something To Read: Galton Boards


I do need to take another light week of writing I’m afraid. There’ll be the Theorem Thursday post and all that. But today I’d like to point over to Gaurish4Math’s WordPress Blog, and a discussion of the Galton Board. I’m not familiar with it by that name, but it is a very familiar concept. You see it as Plinko boards on The Price Is Right and as a Boardwalk or amusement-park game. Set an array of pins on a vertical board and drop a ball or a round chip or something that can spin around freely on it. Where will it fall?

It’s random luck, it seems. At least it is incredibly hard to predict where, underneath all the pins, the ball will come to rest. Some of that is ignorance: we just don’t know the weight distribution of the ball, the exact way it’s dropped, the precise spacing of pins well enough to predict it all. We don’t care enough to do that. But some of it is real randomness. Ideally we make the ball bounce so many times that however well we estimated its drop, the tiny discrepancy between where the ball is and where we predict it is, and where it is going and where we predict it is going, will grow larger than the Plinko board and our prediction will be meaningless.

(I am not sure that this literally happens. It is possible, though. It seems more likely the more rows of pins there are on the board. But I don’t know how tall a board really needs to be to be a chaotic system, deterministic but unpredictable.)

But here is the wonder. We cannot predict what any ball will do. But we can predict something about what every ball will do, if we have enormously many of them. Gaurish writes some about the logic of why that is, and the theorems in probability that tell us why that should be so.

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Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there.

2 thoughts on “Something To Read: Galton Boards”

  1. Thanks for pointing to my blog post. I would like to quote Tim Gowers (A very short introduction to Mathematics, pp. 6) regarding the classical die throwing experiment (“the model”) of probability theory:
    “One might object to this model on the grounds that the dice, when rolled, are obeying Newton’s laws, at least to a very high degree of precision, so the way they land is anything but random…”

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    1. Quite welcome. I’m happy to pass along interesting writing.

      Granted that falling dice, or balls in a Plinko board like this, are moving deterministically. I do wonder if we get to chaotic behavior, in which the toss is nevertheless random. I’m not well-versed enough in the mechanics of this sort of problem to be really sure about my answer. For the balls falling off pins I would imagine that something like twenty rebounds, on either pin or other balls, would be enough to effectively randomize the result.

      (If each rebound doubles the discrepancy between the direction of the ball’s actual velocity and our representation of its direction, then after twenty rebounds the error is about a million times what it started as, and it seems hard to know the direction of a ball’s travel to within a millionth of two-pi radians. But that’s a very rough argument, supposing that randomizing the direction of travel is all we need to have a random ball drop. And maybe two-pi-over-a-million radians is a reasonable precision; maybe we need thirty rebounds, or forty, to be quite sure.)

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