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