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Starting the Scrabble Game (Posted on 2018-01-24) Difficulty: 2 of 5
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).

See The Solution Submitted by Charlie    
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Solution 3 people solution | Comment 7 of 8 |
The following should give the numerator of the probability for a tie with 3 people.  The denominator is simply 100*99*98.  Explanation follows

blank 2*1*98*3
A: 9*8*(7+89*3)
B: 2*1*87*3
C: 2*1*85*3
D: 4*3*(2+81*3)
E: 12*11*(10+69*3)
F: 2*1*67*3
G: 3*2*(1+64*3)
H: 2*1*62*3
I: 9*8*(7+53*3)
L: 4*3*(2+47*3)
M: 2*1*45*3
N: 6*5*(4+39*3)
O: 8*7*(6+31*3)
P: 2*1*29*3
R: 6*5*(4+22*3)
S: 4*2*(2+18*3)
T: 6*5*(4+12*3)
U: 4*3*(2+8*3)
V: 2*1*6
W: 2*1*4
Y: 2*1*1

Letters that occur once can't have a tie and are left out.  
Letters that occur twice, for example, P.  Two people would have to draw P and the third would have to draw one of the 29 tiles that are R-Z.  There are 3 orders.
Letters that occur more than twice, for example, R.  There could be a three-way tie: 6*5*4 or like the previous example 6*5*22*3.  I combined them.
Total 82232/970200 = .085
This is an increase from the two-player solution, I think because the letter distribution is a heavier toward the beginning of the alphabet.

There's no way 100 people could play, but if 100 people draw there's 100% chance of a tie.

  Posted by Jer on 2018-01-25 22:16:50
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