The probability that I am missing only one tile

I recently became fascinated (really!) in this probability problem. I have done quite a bit of work to figure out the answer myself--which I am willing to share with anyone if that will help. But I haven't got an answer yet and I am also just wondering if there is anyone else out there who finds this problem interesting. Or maybe there is someone who knows instantly how to solve it. Anyway, I posted the problem recently on stats.stackexchange but didn't find anyone there interested. I am not a statistician or a student. I am just someone who encountered a problem in real life, started thinking about it and working on it, haven't solved it yet, and now can't get it out of my head.

The problem
The other day I played a word game called bananagrams. Like scrabble it has tiles that represent letters of the alphabet. All letters are represented in the game though not equally. We played with a larger group than normal so I combined two sets. When the game was over I divided the sets but found that 1 letter could not be divided evenly.

There were 5 c's and 283 letters in total. Obviously I was missing at least one tile. What are the chances I am only missing one tile?

Assume (1) any possible number of tiles to begin with is equally likely in the game; (2) the probability of losing each tile is independent of the others and; (3) since we don't know what the probability of losing a tile is, any probability is equally likely; (4) the following distribution. When I go to divide the tiles the letter most represented is e which has 36 tiles. Five letters tie for least represented with four instances each. We can assume the distribution of the others is whatever you'd like (though not exceeding 36 or less than four).

Why I find this question fascinating
I really like the idea of just looking at the end state and while making as few assumptions as possible coming up with a numerical answer about the chance of a certain beginning state. It is something we do intuitively but in trying to solve the problem numerically it seems difficult. Also when I raised this question with friends they thought maybe it was impossible to solve. That eggs me on as it doesn't feel impossible.