Error Rate Calculations

The purpose of this paper is to analyze the following question.

Given an record that where the original no longer exists, but some number of copies have been made of the original, what is the likelihood that the original will be preserved somewhere in the first generation copies?

An analysis of this question can be made from the field of probability and statistics.

ASSUMPTIONS

1. Assume that the error rate is completely random. This is not likely since some types of errors are more likely than others. These types errors are well documented.
2. Only the first generation copies are analyzed in this paper.

VARIABLES USED

• M is the number of characters in the record
• N is the number of copies of the original record
• P is the error rate in the copies

EQUATIONS

•  
• M*P is the number of errors per copy
• M*N*P is the total number of errors
• PN is the probability that all copies will have an error in a particular character
• M*(PN) is the probability that all copies will have an error in any same character

EXAMPLES

Example #1:

   M=100,000 characters
   N=10 copies made of the original record
   P=.001 error rate (0.1%)

1. 100 errors per copy
2. 1000 total errors
3. .00110 = 1x10-31
4. (1x105) x (1x10-31) = 1x10-26

Example #2:

   M=100,000 characters
   N=2 copies made of the original record
   P=.001 error rate (0.1%)

• 100 errors per copy
• 1000 total errors
• .0012 = 10-6
• 105 x 10-6 = 1/10

ANALYSIS OF EXAMPLES

1. Example #1 shows a "best case" situation. In this example the original had 10 copies made. The probability is very small that there are multiple errors in the same character. The probability, although small, is not equal to zero.
2. Example #2 shows a one in ten chance that the original was lost for a single character. The chance that more than one character from the original was permanently lost is much smaller. One single character out of a record of over 100,000 is not a lot of error.

CONCLUSIONS

If there was only one copy of the original made and there were any errors in the copy, there is no way that the original can be restored, as it is permanently lost. The worst case scenario still shows a very small error rate.

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Last Updated 1-1-97