Is Scrabble's method of determining turn order fair? At least the way my family plays, turn order is determined by drawing tiles and seeing who has the letter closest to A (blanks taking precedence). If I recall correctly, there are 100 unevenly distributed letters.
Intuitively, it seems like it would be unfair, though I can't come up with a way to prove it. Obviously the distribution matters: my thoughts are if there are more tiles in the first half of the alphabet (including blanks), then the starting player holds an advantage since they're more likely to get a tile earlier in the alphabet than the next (and vice versa if there are fewer tiles in the first half). That doesn't seem too right, though. I'm probably just forgetting some basic prob stats.
I imagine this is likely a duplicate, but I couldn't find anything relating to it (maybe since I've been searching for Scrabble). My apologies if it is.
Please feel free to edit in appropriate tags.

Look at a toy problem. Suppose there are two players and three tiles: $A_{1}$, $A_{2}$, and $B$. The subscripts on the $A$,s indicate there are two $A$ tiles. Look at all the possible outcomes: the first letter is what the first player draws, the second letter is what the second player draws and the third letter is the remaining letter.


*

*$A_{1}A_{2}B$

*$A_{1}BA_{2}$

*$A_{2}A_{1}B$

*$A_{2}BA_{1}$

*$BA_{1}A_{2}$

*$BA_{2}A_{1}$


The first and third outcomes are draws. The first player wins the second and fourth outcomes. The second player wins the fifth and sixth outcomes. In this example; if the first player draws a good tile, an $A$, the first player can do no worse than draw; if the first player draws a bad tile, a $B$, the second player has to win.
