Even odds

This evening, I found a nice probabilistic puzzle on http://www.futilitycloset.com/“A bag contains 16 billiard balls, some white and some black. You draw two balls at the same time. It is equally likely that the two will be the same color as different colors. What is the proportion of colors within the bag?”
To be honest, I did not understood the answer on the blog, but if we write it down, we want to solve

http://freakonometrics.blog.free.fr/public/perso5/futil-01.gif

Let us count: if http://freakonometrics.blog.free.fr/public/perso5/futil-04.gif is the total number of balls, and if http://freakonometrics.blog.free.fr/public/perso5/futil-03.gif is the number of white http://freakonometrics.blog.free.fr/public/perso5/futil-05.gifballs then

http://freakonometrics.blog.free.fr/public/perso5/cccccwwwwwww.gif

I.e. we want to solve a polynomial equation (of order 2) in http://freakonometrics.blog.free.fr/public/perso5/futil-07.gif, or to be more precise, in  http://freakonometrics.blog.free.fr/public/perso5/futil-00.gif

http://freakonometrics.blog.free.fr/public/perso5/futil-06.gif

If http://freakonometrics.blog.free.fr/public/perso5/futil-04.gif is equal to 16, then http://freakonometrics.blog.free.fr/public/perso5/futil-03.gif is either 6 or 10. It can be visualized below

> balls=function(n=16){
+ NB=rep(NA,n)
+ for(k in 2:(n-2)){
+ NB[k]=(k*(k-1)+(n-k)*(n-k-1))
+ }
+ k=which(NB==n*(n-1)/2)
+ if(length(k)>0){
+ plot(1:n,NB,type="b")
+ abline(h=n*(n-1)/2,col="red")
+ points((1:n)[k],NB[k],pch=19,col="red")}
+ return((1:n)[k])}
> balls()
[1]  6 10

But more generally, we can seek other http://freakonometrics.blog.free.fr/public/perso5/futil-04.gif‘s and other pairs of solutions of such a problem. I am not good in arithmetic, so let us run some codes. And what we get is quite nice: if http://freakonometrics.blog.free.fr/public/perso5/futil-10.gif admits a pair of solutions, then http://freakonometrics.blog.free.fr/public/perso5/futil-10.gif is the squared of another integer, say http://freakonometrics.blog.free.fr/public/perso5/futil-13.gif. Further, the difference between http://freakonometrics.blog.free.fr/public/perso5/futil-11.gif and http://freakonometrics.blog.free.fr/public/perso5/futil-12.gif is precisely http://freakonometrics.blog.free.fr/public/perso5/futil-13.gif. And http://freakonometrics.blog.free.fr/public/perso5/futil-12.gif will be one of the answers when the total number of balls will be http://freakonometrics.blog.free.fr/public/perso5/futil-20.gif. Thus, recursively, it is extremely simple to get all possible answers. Below, we have http://freakonometrics.blog.free.fr/public/perso5/futil-04.gifhttp://freakonometrics.blog.free.fr/public/perso5/futil-03.gifhttp://freakonometrics.blog.free.fr/public/perso5/futil-21.gif and the difference between http://freakonometrics.blog.free.fr/public/perso5/futil-21.gif and http://freakonometrics.blog.free.fr/public/perso5/futil-03.gif,

> for(s in 4:1000){
+ b=balls(s)
+ if(length(b)>0) print(c(s,b,diff(b)))
+ }
[1] 9 3 6 3
[1] 16  6 10  4
[1] 25 10 15  5
[1] 36 15 21  6
[1] 49 21 28  7
[1] 64 28 36  8
[1] 81 36 45  9
[1] 100  45  55  10
[1] 121  55  66  11
[1] 144  66  78  12
[1] 169  78  91  13
[1] 196  91 105  14
[1] 225 105 120  15
[1] 256 120 136  16
[1] 289 136 153  17
[1] 324 153 171  18
[1] 361 171 190  19
[1] 400 190 210  20
[1] 441 210 231  21
[1] 484 231 253  22
[1] 529 253 276  23
[1] 576 276 300  24
[1] 625 300 325  25
[1] 676 325 351  26
[1] 729 351 378  27
[1] 784 378 406  28
[1] 841 406 435  29
[1] 900 435 465  30
[1] 961 465 496  31

Thus, given http://freakonometrics.blog.free.fr/public/perso5/futil-22.gif, consider an urn with http://freakonometrics.blog.free.fr/public/perso5/futil-23.gif balls. We draw two balls at the same time. It is equally likely that the two will be the same color as different colors. Then the number of colors within the bag are respectively

http://freakonometrics.blog.free.fr/public/perso5/futil-24.gif

Finally, observe that the http://freakonometrics.blog.free.fr/public/perso5/futil-11.gif‘s are well known, from Pascal’s triangle,

also known as triangular numbers,

http://freakonometrics.blog.free.fr/public/perso5/tr-pascal.gif

Maths can be magic, sometimes…


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