Problem:
With over 7,000 words in the Xsketch Repertoire, it is common to find the same word repeated 5 times in about a 4 hour period. Must we presume the word algorithm is only pseudo-random, or are these probabilities to be expected?
Solution:
Let’s first get an idea of the average length of time per sketch. For the sake of simplicity, I shall concern myself with the averages of this past month July.
Here is a selection of 20 Random Average Sketch Times (of people who play Xsketch often and inhabit the higher tables):
49 s, 41 s, 40 s, 43 s, 30 s, 38 s, 44 s, 52 s, 25 s, 46 s, 48 s, 46 s, 44 s, 44 s, 42 s, 46 s, 35 s, 58 s, 41 s, 47 s.
The average of the above is 860/20 = 43 Seconds.
43 seconds is the same as 0.71666 of a minute.
In 4 hours, which is 240 Minutes, there are approximately 335 sketches.
Now, the probability that a specific word will show up on any one turn is about 1/7000. In order to find the probability of five such occurrences in 335 trials, we turn to the binomial distribution.


P(x=5)=(335¦5) (1/7000)^5 (6999/7000)^330

(335¦5)=335!/(5!*330!)= 12370110840/24=34120889067
Multiplying by the other fractions…
P(x=5) = 1.93666629 × 10-9
Quite a small number, but let us not forget we have 7000 such numbers.

Now, the probability that a specific word won’t repeat five times is
1-1.93666629 × 10-9
With 7,000 words, the probability that none of them will repeat is
(1-1.93666629 × 10-9)^7000 = 0.999986443
The probability that at least one will repeat four times is
1 - 0.999986443= 1.3557 × 10-5
This is the equivalent of 0.0013557%
Very unlikely.
I am forced to conclude that the Xsketch algorithm is only pseudo-random, although it is very likely my mathematics may be flawed.