# What are the odds of reaching the end of a random choice path without repeat choices?

Given $m$ balls in a container, having $n\le m$ colors, if balls are chosen from the container randomly without replacement until all balls have been removed and the order of choices is considered, what are the odds that a given "choice path" will contain no instances of two or more balls of the same color chosen consecutively?

By extension, given a number of colors $n$, what is the largest number of balls $m$ such that the odds of a random "choice path" containing no repeat choices is greater than $x$% (assuming that the colors are spread as evenly as possible, i.e., the count of balls for any particular color is no more than $1$ greater than the count of balls for any other color)?

• How many balls of each color do we have? Suppose $n=2$, $4$ red balls, $1$ blue ball. – hhsaffar Dec 14 '13 at 6:19
• The "extension" comment is intended to limit the "color spread" in the question. Basically, no color should mark any proportion of the balls greater than any other color by any significant amount, or more directly, the colors should be evenly spread. – abiessu Dec 14 '13 at 6:23