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.)
Answer:
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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