A book titled The Bible Code introduced the topic of equidistant letter sequences (ELS), described below, for finding words “hidden” in text. That book referenced the Hebrew Bible, but prompts a question about finding any given word in any, say, English-language text.

For simplicity, and to better match the Hebrew, spaces and punctuation are removed. A particular text that I have in mind, thus crunched, has 284,939 characters remaining (letters and digits). How many times would you expect to find the word FLOOBLE as an equidistant letter sequence in the text? Ignore case. The word can start at any of the 284,939 characters and proceed by skipping any constant number of letters forward or backward. So, for example, if the 11,000th character were an F and the 10,000th an L, and the 9,000th an O, etc. that would be one occurrence. Of course we don’t expect always to find such decimally round spacings. The question again, How many do we expect to find?

The absolute and relative frequencies of the relevant letters in the text are:

B  4771 0.016744
E 36232 0.127157
F  7167 0.025153
L  9563 0.033562
O 22486 0.078915

that is, for each letter is shown the number of occurrences in the text and that number divided by the total of characters in the text.

(In reply to remaining solution by Cory Taylor)

If you flip a coin once, the expected number of heads is 1/2. In larger number of trials with smaller probability, the expected value is merely something to plug into the Poisson formula to get probabilities of various integer amounts. If the .85 were correct, it would mean only that there's a 43% chance of none, a 36% chance of 1, a 15% chance of 2, a 4% chance of 3, a 1% chance of 4, and diminishing chances above that.

I do think people are making it harder than it is. Try concentrating, not on the starting position and skip value, but rather on the starting position and the ending position, and the restriction on the ending position given any particular starting position.

Posted by Charlie on 2003-03-21 03:36:56