Activity 3: Coding Errors
(Teacher Copy)

Students will

1. use problem-solving skills to solve a problem involving with modular arithmetic (mod 10),
2. find the probability of missing a coding error, and
3. investigate the base-2 number system.

Introduction:

• To investigate one of the errors that can occur in a bar code system.
• To investigate the probability that a transposition error will not be detected.
• To investigate the binary number system.

Part A

1. Ordered pairs that go undetected are: (0,5); (1,6); (2,7);...(9,4)--a total of 10 pairs.  There are many different 2-digit numbers that the students will have to consider (10 numbers to choose from for each of the two digits, making a total of 10 x 10 or 100 permutations of numbers).  However, groups will not even have to consider the possibility of an error of transposition if the two digits are identical.  Thus, the number of possible permutations of numbers that have an opportunity to be transposed is 10 x 9 = 90.  These pairs all fit the equation |a - b| = 5

   Because of the differences in learning styles of students some groups may approach this problem in a very disorganized looking trial-and-error format, some may use an organized list or table, and some may try to use a mod 10 congruence statement to find the kinds of numbers that would give the same mod 10 value despite being transposed.

2. There are 10 numbers to choose from for each of the two digits, making a total of 10 x 10 or 100 permutations of numbers for the two digits in question, with one exception.  There is no possibility of an error of transposition if the two digits are identical.  Thus, the number of possible permutations of numbers that have an opportunity to be transposed is 10 x 9 = 90.  For each number "a" that is chosen, a unique number "b" can be chosen that satisfies the absolute value condition.  Ordered pairs that go undetected are: (0,5); (1,6); (2,7);...(9,4)--a total of 10 pairs.  Therefore, the probability of an error of transposition in the UPC-A system is 10/90 or 1/9 or 11.1%.  The IBM system has problems only with ordered pairs (0,9) and (9,0).  Thus, the probability of an error not being detected is 2/90 or 2.2%.

Discussion

Have students discover the complement rule of probability by asking them to find the success rate for detecting errors of transposition.  See R1, page 1.

What do you notice about the pattern of the answers to Part B?  Students might observe that the two numbers in base 10 for each encoding add to 127 (base 10).  They might notice that there are always a total of 7 "ones" and 7 "zeros" for each code.  Students might see that between the two binary numbers in the left and right parity there is always one-and-only-one "one" in each of the seven base 2 places.  Thus, the total for the two numbers in base 10 would necessarily consist of 1 sixty-four, 1 thirty-two, 1 sixteen, 1 eight, 1 four, 1 two, and 1 one.