A statistical analysis: distributing pre-made characters

Last post, I described the problem of distributing pre-made characters to a group of picky roleplayers. Players get to choose or are assigned a character. If they are assigned a character, they may not like it. Let's see what method of distribution works the best.

Let's have ten players and ten pre-made characters. Each player would be satisfied, for example,  playing 5 of those characters. The odds of a player getting a character he or she enjoys in a random draw would be 0.5. The odds that all players would get a character they enjoy would be 1024 to 1 (0.098%). Consider it like ten coin tosses, and we calculate the odds of getting ten heads in ten tosses.  It would seem fair, since no one person or group of people would have preferential treatment, but odds are, one or more people would not like their character.

Now, imagine each player chose a character, one by one, until they were all gone. Let's not think about how we decide what order to have players choose in. The odds of each person having a character they would enjoy shrinks as the pool of characters shrink. However, the first five picks would have a chance of 100% that they'd find a character they like, because each player likes 5 characters, and less than 5 characters have been chosen, guaranteeing at least one they enjoy. This creates a better opportunity for those who get to choose first, however, it creates the air of unfairness for those who have to choose last.

Enjoyment (all) = 1 * 1 * 1 * 1 * 1 * 0.5 * 0.4 * 0.3 * 0.2 * 0.1 = 0.12% (833.33 to 1)

Now imagine a random draw + trading option. Half of the people will get something they like, half will not. Of those that do not, they are allowed to trade characters. 2 pairs of players could trade if the other person had something they liked. Therefore, the odds of satisfying every player is zero if we have an odd number of unsatisfied players. But what are the odds of either two, four, six or eight players having a character they do not like? Again, we use the coin toss analogy:

Odds of an even number of heads on a coin toss = 0.205 + 0.205 + 0.044 + 0.044 = 0.498

So, it's almost a 50% chance that there are an even number of characters that can be traded. The math seeing if two players can trade is pretty complex, so I will skip it. Suffice to say that it's a greater than 50% chance one or more people will be unhappy.

What about letting players choose the five characters beforehand, and the GM (me) doling out who gets what? What are the odds of each character getting chosen by at least one player? The odds of any of the ten characters not getting chosen by a player is 1024 to 1 (0.097%). This sounds like the best option.

Obviously, there are many confounding issues to the analysis. Maybe players are more or less picky. Maybe some characters seem really fun to play, and some are so boring or unimportant no one wants to play them. Maybe there are some players who would be happy playing anything.

Hopefully you've enjoyed my crappy math. Feel free to correct mistakes.