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Expected Number of ELSs

In order to perform a Torah code experiment, the minimum skip and maximum skip allowed for an ELS of a given key word must be specified. One of the ways of specifying this is to set the minimum skip to 1 or 2 and the maximum skip so that the expected number of ELSs is equal to a user specified value.

Expected number of ELSs is just the average number of ELSs to be produced in a given monkey text population. The monkey text population used for this purpose is the control population of random letter permuted texts. Suppose that the given key word is טבח. What should the largest allowed skip for an ELS of this word? Since the Torah code hypothesis is that the effect occurs with the smaller skip ELSs, we must have a criteria for smaller skip. But smaller skip cannot mean smaller skips than say 5, regardless of the given key word. Key words that have many letters or less frequently occurring letters will most likely not have any ELSs with skips smaller than 5, while key words that have few letters with no infrequently occurring letters will most likely have hundreds of ELSs.

This problem is handled by setting the maximum skip permitted for an ELS of a given key word to be set so that the expected number of ELSs produced by a text from the monkey text population is equal to the user specified value.

Expected number means average. So to understand this, suppose that the computer search procedure looking for ELSs finds all ELSs of the word טבח between skips 2 and S_max. Each time the computer search selects at random a text from the monkey text population and searches for all ELSs of the word טבח, such that the ELS has skip between 2 and S_max. Each time the search is executed on a text, a generally different number of ELSs will be found. The average value of the number of ELSs found taken over all the texts of the monkey text population is the expected number of ELSs. For the word טבח, if the smallest allowed skip is 2 and the maximum allowed skip is 23, then the expected number of ELSs found in the randomly letter permuted monkey text population will be about 100.

More formally, given a key word w, a minimum absolute skip S_min and a maximum absolute skip S_max, we associate with each text T in the letter permuted population the set E(w,T,S_min,S_max). This set is the set of all ELSs of word w in the text T that have absolute skips in the interval [S_min,S_max]. This set has a size: the number of ELSs it contains. The arithmetic average of the sizes of the ELS sets taken over all the texts of the population is defined as the expected number of ELSs