# Talk:Bean machine

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## Pic

A picture of a bean machine would make this article much clearer. Michael Hardy 02:10, 16 May 2004 (UTC)

(Obviously, pictures have been added since Michael posted his comment. - dcljr (talk) 07:19, 20 October 2015 (UTC))

## But how does it work?

Can anyone explain why it aproximates a bell curve or normal distribution? The article doesnt define why it just isnt random. —The preceding unsigned comment was added by 159.153.156.60 (talk) 10:30, 3 May 2007 (UTC).

Give Binomial distribution a read and see if it answers your question. Essentally, each peg in the bean machine/Galton board presents a left/right binary decision and the resulting bin that the the beans/balls eventually fall into is a result of a series of these left/right binary decisions.
Atlant 12:07, 3 May 2007 (UTC)

The statement

"According to the central limit theorem the binomial distribution approximates normal distribution provided that n, the number of rows of pins in the machine, is large."

is incorrect. The central limit theorem is freqently misapplied in this fashion. It has to do with the distribution of the SUM of the random variables. This does not mean that if you "sum" enough graphs of the binomial distribution that it will bcome a normal distribution. —Preceding unsigned comment added by 192.25.240.225 (talk) 15:13, 31 March 2008 (UTC)

does that statement really wrong? I do not think so. As I knew, when a ball is falling down from the top of the machine, it will bounce left and right as they hit each pin. And each hit is an binomial experiment. The final position a ball get depends on the SUM of the results of the experiments(hits). Then, the height curve of the balls is the Probability density function curve of binomial distribution. according to CLT, it will be a bell curve.Chaosconst (talk) 06:35, 16 September 2008 (UTC)
It's not wrong. Technically, the de Moivre–Laplace theorem gives the result, but since that theorem is a special case of the CLT, saying the CLT gives the result is just as correct. - dcljr (talk) 07:19, 20 October 2015 (UTC)

## Clarity of machine description

I had not come across Galton's box before. It would have been helpful to point out that balls are fed to the machine centrally at the top.

Tony.payton (talk) 10:41, 15 September 2008 (UTC)

Actually, given the wide "bowl" (or "twin ramps") shape at the top of Galton's diagram and then the narrower "bowl" below, you should be able to feed in the bean anywhere at the top, not just in the very center. (Unless you just meant that it's not fed in from the side of the box. In which case, yeah.) But come to think of it, he probably did that so he could pour in lots of beans at once (in which case the beans start interfering with each other's movements, and the motions are no longer independent). Perhaps I'm overthinking this… - dcljr (talk) 07:53, 20 October 2015 (UTC)

## Distribution of the balls

The article states "If a ball bounces to the right k times on its way down (and to the left on the remaining pins) it ends up in the kth bin counting from the left." Surely this cannot be correct? If the ball bounces right k times and left l times it will end up (k - l) bins to the right of the central bin?

Tony.payton (talk) 10:47, 15 September 2008 (UTC)

In the second picture (photograph), if a ball never goes to the right, you can see it would end up in the "zeroth" (leftmost) bin; if it goes to the right every time, it would end up in the last (rightmost) bin. You don't need to keep track of both left and right movements. The top picture (drawing) suffers from the fact that a ball can actually reach the left or right side of the box and get bounced back towards the center (thus every once in a while, the sequence of random movements is interrupted by a non-random movement towards the center). If the box were made wider, the balls could be prevented from hitting the sides of the box, avoiding that problem. - dcljr (talk) 07:19, 20 October 2015 (UTC)

## Three dimensional bean machine

I am preparing a paper in which I use the bivariate binomial distribution proposed by Aitken and Gonin (1935), which is based on a fourfold sampling procedure in contrast to the twofold sampling procedure in the ordinary bean machine. For a demonstration one actually needs a three dimensional bean machine. In the traditional bean machine, after the ball has fallen on the first pin, the ball can fall on one of two pins with probabilities p1 and p2 with p1+p2 = 1. In the case of a fourfold sampling procedure according to Aitken and Gonin after the ball has fallen on the first pin, the ball can fall on one of four pins, arranged as the corner points of a square, with probabilities p1, p2, p3, p4 with p1+p2+p3+p4 = 1 and so on. Does one know of such three dimensional bean machine.

Reference

Aitken, A.C. and Gonin, H.T. On fourfold sampling with and without replacement, Proc. Roy. Soc. Edinburgh 55, 114–125, 1935. — Preceding unsigned comment added by Ad van der Ven (talkcontribs) 18:51, 30 January 2012 (UTC)