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. Assume the distribution of the others is whatever you'd like (though not exceeding 36 or less than four).

Simplify the problem if you'd like as long as the key difficulties are still solved for.

I have found this problem strangely interesting to me. I have done some work to try to solve this myself but am not there yet. I'd be happy to share my work so far if someone wants to look it over but wonder if it wouldn't be better to start by getting your thoughts on the correct approach to take in case I may be headed in completely the wrong direction.

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    $\begingroup$ While it's a potentially interesting probability problem, some problems are much more easily solved by other means. [It's a trivial matter to google bananagrams letter tiles distribution and find out there are 144 tiles in the game - and how many of each there are. You are missing 5 tiles total. Apparently they will replace up to three tiles from a game.] $\endgroup$
    – Glen_b
    Nov 12, 2016 at 8:50
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    $\begingroup$ Thanks. Agree that is a lot easier. In reality, I think it is more likely that I miscounted the number of tiles on the table in putting together this problem than that I really lost 5 tiles. But taking the assumptions of the problem for what they are I am still interested in figuring out how to answer it if anyone can help with that. $\endgroup$
    – snowguy
    Nov 12, 2016 at 9:03


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