Possible anomaly in Cryptonomicon

Question

Toward the beginning of Neal Stephenson's Cryptonomicon, Avi gives Randy two strings of sixteen supposedly random hexadecimal numbers as encryption keys:

AF  10  06  E9  99  BA  11  07  64  C1  89  E3  40  8C  72  55

67  81  A4  AE  FF  40  25  9B  43  0E  29  8D  56  60  E3  2F

If you change the hexadecimals to decimals and change the decimals to letters, modulo 25 without J, like Lawrence does with his one-time-pads later in the book, you get:

z   k   f   h   c   l   r   g   z   s   m   b   o   p   o   k

c   d   o   y   m   o   m   e   r   o   q   q   l   v   b   w

The first sequence contains two Os and the second contains three Os. The probability of randomly picking at least five Os out of twenty-five letters given thirty-two picks is less than 1%, which makes it seem like the numbers are not actually random.

Additionally, there are six prime numbers in each sequence:

E9  11  07  C1  89  E3

67  25  43  29  E3  2F

(all primes in base sixteen)

What does this mean?

Answer 1

I admit that I don't have the math to calculate the odds precisely, but we should start out by observing that the letter "O" is not particularly interesting here. What we're concerned with is the chance of some letter being repeated 5 times in a string of 32 draws. What are the odds of that not happening? I suspect they're not very good.

Let's consider a slightly simpler case: what are the odds of drawing 25 tiles from a 25-tile set, with replacement, and getting each tile represented? Clearly it's 24!/(25^25), right? The first tile is free, we have a 1/1 chance of getting a unique item. Then we have 24/25 odds of a unique second tile, times 23/25 odds for a unique third tile, and so on down to 1/25 for the last. So we're looking at 25/25 * 24/25 * ... * 2/25 * 1/25, which is a very small number.

Someone with a better training in probability than I've got will have to extend this to the problem of drawing 32 with not more than 5 repeats, but the following little bit of python consistently finds ~.17 percent chance of getting 5 or more repeats when drawing 32 times with replacement from a bag of 25.

>>> def trials(n=10000):
...   count = 0
...   for i in range(n):
...     c = Counter([randint(1,26) for _ in range (32)])
...     if 5 <= max(c.values()):
...       count += 1
...   return count/float(n)
... 

So a little less than 1/5, which is not such bad odds. So I think the premise of the question, that the repetition of some letter 5 times in a sequence of 32 draws with replacement from a set of 25 is highly unlikely, is incorrect, and this is not an anomaly.