Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have $n=100$ balls, in which $h=3$ are red, 97 are blue.

I randomly place the balls in a circle, then check the minimum interval of red balls (e.g., if 2 red balls are consecutive, then their interval is 1).

Now I am interested in the distribution of this statistic, or at least the mode.

share|improve this question
2  
As long as $h=3$, it seems to be easy to simply count the number of configurations with each possible minimum. There will be one term for when exactly one pair of balls has the given distance, and one for when a central red ball has the minimum distance to both of its partners. The mode is 1 and the distribution is one-sided triangular. –  Henning Makholm Aug 31 '11 at 19:57

1 Answer 1

up vote 5 down vote accepted

Consider the following experiment: randomly place the balls in the way you mention, choose one of the red balls randomly, and tag it as "first". From the point of view of the first ball, the other two red balls are placed randomly.

Denote by $x,y,z$ the intervals between the red balls, starting with the first and going clockwise. So $x+y+z = 97$, and $x,y,z \geq 0$ are integers. All possible triples $x,y,z$ satisfying these conditions are equiprobable. It is now possible to obtain exactly the entire distribution by enumeration of the $\binom{99}{2}$ cases.

Here is the entire distribution (multiplied by $\binom{99}{2} = 4851$): $$ \begin{array}{ccccccccccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ 291 & 282 & 273 & 264 & 255 & 246 & 237 & 228 & 219 & 210 & 201 \\ \\ 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 \\ 192 & 183 & 174 & 165 & 156 & 147 & 138 & 129 & 120 & 111 & 102 \\ \\ 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30 & 31 & 32 & 33 \\ 93 & 84 & 75 & 66 & 57 & 48 & 39 & 30 & 21 & 12 & 3 \end{array} $$ The interested reader can easily prove the formula $$h(x) = 300 - 9x.$$

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.