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).
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.
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Posted by Jer
on 2018-01-25 22:16:50 |
3 people solution
