# what's the most probable min interval in 100 balls in a circle?

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.

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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

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 $$h(x) = 300 - 9x.$$