Bean machine (Galton board) explanation

Question

I am confused by the marked part of the following explanation (see below; https://en.wikipedia.org/wiki/Bean_machine). Suppose the bead bounces to the right twice and to the left also twice. It will land exactly at the center. But according to this explanation it will land to second bin counting from the left. It is also not clear to me, whether by "counting from the left" they mean from the most left or from the center. For similar explanation, see: http://mathworld.wolfram.com/GaltonBoard.html

Can you please reconcile this to me?

Thanks!

Answer 1

It´s a matter of definition how someone counts the bins. In the article of wiki they start with the number zero: $0,1,2,3,...,n$

Quote:

the number of paths to the $\color{blue}k$th bin on the bottom is given by the binomial coefficient ${n\choose \color{blue}k}$.

Therefore the first left bin is denoted by $k=0$. Consequently in the case of $n=4$ the second bin is indeed the center: $01\color{red}234$

Answer 2

There is no contradiction.

• If it never bounces to the right, it lands in the left most bin.

• If it once bounces to the right, it lands one to the right of the left most bin.

• If it twice bounces to the right, it lands two to the right of the left most bin.

And so on.

In your example there are a total of four times it bounces, thus there are five possible final position. And "two to the right of the left most bin" indeed is right in the middle.