To determine who plays first in a game of Scrabble, each player draws a tile and the one closest to A wins, except if one draws a blank tile. A blank tile beats any letter.
If there are only two players, what is the probability they both draw the same letter or both draw a blank?
The letters (and counts) are
E (12); A, I (9); O (8); N, R, T (6); D, L, S, U (4); G (3); B, C, F, H, M, P, V, W, Y (2); J, K, Q, X, Z (1); and 2 blanks.  100 tiles in all.
Bonus: What if there are more than two players: what is the probability of a tie for who plays first? (subsequent play is to the left so 2nd, etc. player are not determined by the draw).
For a letter that occurs n times, the probability that both draw it is n/100*(n1)/99. Each letter is independent so we can just add them up:
(12*11+9*8*2+8*7+6*5*3+4*3*4+3*2+2*1*10+1*0*5)/(100*99)
=496/9900=.05010101
The bonus is pretty complicated. I'll need to save it for later.
Edit to fix errors. I made an impressive 3 total.
Forget to consider the blank. (9 changed to 10)
Had a + that should have been * (2*1+9, now 2*1*10)
Wrong denominator. Draws without replacement (10000 changed to 100*99)
Edited on January 25, 2018, 3:07 pm
Edited on January 26, 2018, 9:40 am

Posted by Jer
on 20180124 14:47:34 