Question

A Scrabble game has identical copies of each letter tile numbering as follows: A-9, B-2, C-2, D-4, E-12, F-2, G-3, H-2, I-9, J-1, K-1, L-4, M-2, N-6, O-8, P-2, Q-1, R-6, S-4, T-6, U-4, V-2, W-2, X-1, Y-2, Z-1 and Blanks-2. Seven tiles are randomly sampled without replacement from the above 100 tiles.If a letter appears exactly once in this seven-tile sample, we call it a singleton in this problem.

a. Consider a letter that has n copies in total (e.g., for the letter E, n=12). What is the probability that this letter is a singleton?

b. What is the expected value of the number of singletons? (Count “blank” as a letter.)

Answer 1

Suppose there are n copies of a letter 'x',

Then the probability of tile being the specific letter 'x' = n/Total sample size = n/100.

Since every tile has the probability of being letter 'x' = n/100. This is an example of bernoulli trial.

Thus the probability of finding 1 letter in the sample size 7 follows a binomial distribtuion,

The probability is given by :

The probability of letter being a singleton is

=

b.)

The probability value for each letter is as given below:

Letter	Success probability (p)	(1-p)	Probability of singleton
A	0.09	0.91	0.357758
b	0.02	0.98	0.124018
c	0.02	0.98	0.124018
d	0.04	0.96	0.219172
e	0.12	0.88	0.390099
f	0.02	0.98	0.124018
g	0.03	0.97	0.174924
h	0.02	0.98	0.124018
i	0.09	0.91	0.357758
j	0.01	0.99	0.065904
k	0.01	0.99	0.065904
l	0.04	0.96	0.219172
m	0.02	0.98	0.124018
n	0.06	0.94	0.289745
o	0.08	0.92	0.339559
p	0.02	0.98	0.124018
q	0.01	0.99	0.065904
r	0.06	0.94	0.289745
s	0.04	0.96	0.219172
t	0.06	0.94	0.289745
u	0.04	0.96	0.219172
v	0.02	0.98	0.124018
w	0.02	0.98	0.124018
x	0.01	0.99	0.065904
y	0.02	0.98	0.124018
z	0.01	0.99	0.065904
Blanks	0.02	0.98	0.124018

Expected value of number of singletons

= sum of probabilities of all the letters being singleton

= 4.93572