# A Scrabble game has identical

A Scrabble game has identical copies of each letter tilenumbering 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 sampledwithout replacement from the above 100 tiles.If a letter appearsexactly once in this seven-tile sample, we call it a singleton inthis problem.

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

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

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 7follows 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 ofsingleton 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

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