SKIP TABLES FOR THE ROFFMAN SKIP FORMULA
ROFFMAN SKIP FORMULA.
The number of ways that a term (either forwards, backwards, or diagonal) can fit into a matrix is determined as follows:
(1) Let the number of skips possible in a forward direction on a row of length (r), where r = the number of columns in the matrix, be equal to "Sr."
(2) Likewise, let the number of skips possible in a vertical direction on a column of length (c), where c = the number of rows in the matrix, be equal to "Sc."
(3) The Roffman Skip Formula for total skips is as follows:
Skips = 2(Sr + Sc + 2[Sr][Sc]) = 2Sr + 2Sc + 4SrSc.
An example of skip value determined through use of the above tables and formula follows: Find the number of skips possible for a 4-letter word in a Matrix 28 columns by 11 rows.
Solution: Skip Tables for words ranging between 3 and 8 letters are found at the bottom of this web page and on pages 215 through 220 in Appendix B to my Ark Code book (there is a slight correction below for words 8 letters long). For a 4-letter word use Table 1B. On it find that 28 columns = 9 possible skips forward. Thus Sr = 9. Now note that 11 rows = 3 possible skips vertically. Thus Sc = 3. Now apply the formula which is Skips = 2(Sr + Sc + 2[Sr][Sc]) 2(9 + 3 + 2(9)[3]) = 2(12 + 54) = 2(66) = 132 SKIPS. Now, let us suppose that the 4-letter term occurred at skip 100. To get an idea of how likely such a term is to be found at an ELS, search a range of 132 skips, such as from skip 101 to skip 232. The number of "hits" for this term is then divided by letters in the Control (if this is scrambled Torah the number of letters in the Control is the same as in Torah, i.e., 304,805). The quotient is the Word Frequency Per Letter. This is multiplied by the number of letters on each matrix to reveal Word Expectancy Per Matrix. It is inherent in this procedure that the larger the number of letters in the matrix, the larger the number of placements possible for any given key word at any ELS. After determining Word Frequency Per Plot we apply the Poisson Equation to see the probability that they are present at least once. This is necessary to determine a true probability for each word. Just because a word is likely to appear once per plot does not imply it will always be there. Words may average out to many times per plot area without actually being in a given plot of that area. Of course, if the expected frequency is sufficiently high we eventually reach a probability like .9999999 which we simply round off as 1.0.
HOW TO FIND THE CHANCE OF A TERM APPEARING AT LEAST ONCE*
1. FIND PROBABILITY IT DOES NOT OCCUR BY POISSON EQUATION.
x (-lambda)
f(x) = Lambda e x = 0; lambda = expected frequency per matrix
x!
2. 1 ‑ f(0) = THE PROBABILITY OF OCCURRING AT LEAST ONCE.
(where f(0) = the probability it will not occur)
3. On an Excel or Works spreadsheet, head columns as follows: A: Whatever identifies the calculation; B: Skips Used on the Matrix, C: Number of hits (on CodeFinder or similar software) in Skip Range; D: Divide by 304,805 Letters in Torah or Control; E: The Quotient Equals Frequency Per letter; F: E Quotient Multipled by Letters on Matrix = Word Expenctancy; G: Poisson Equation = 1-EXP(-F#) where # equals the row number of the item in column F in question on the spreadsheet. If you want to know the chance for the item to be on the matrix, head column H accordingly. The value of column H will be the reciprocal of the value found in column G by Poisson Eqiation.
* Note: While this author (Barry S. Roffman) discovered the Roffman Skip Formula, my eldest son (an MIT graduate), Rabbi Moshe Roffman, is the author of the spreadsheets and the man who first introduced use of the Poisson Equation into my research.
Torah Code Restrictions and Modification to Probability Calculations.
