A problem of parcel tracking numbers.
Carl James Schwarz (P.Stat. (Canada), PStat (US)
Department of Statistics and Actuarial Science
Simon Fraser University
Burnaby, BC, V5A 1S6
cschwarz@stat.sfu.a
This reports deals with various issues about a Canada Post parcel tracking number found
a slip of paper (CP 001 201 740 SY) that was found in possession of a suspect and its
relationship to a tracking number of a parcel of interest (CP 001 201 748 SY). The two
tracking numbers differ in the last digit before the final alphabetic suffix.
Information on the construction of tracking numbers was provided by an employee of
Canada Post and is also available at http://en.wikipedia.org/wiki/Canada_Post
“Canada Post uses a 13 character barcode for their pre-printed labels. Bar codes
consist of two letters, followed by eight sequence digits, and a ninth digit which is
the check digit. The last two characters are the letters CA. The check digit seems
to ignore the letters and only concern itself with the first 8 numeric digits. The
scheme is to multiply each of those 8 digits by a different weighting factor, (8 6 4
2 3 5 9 7). Add up the total of all of these multiplications and divide by 11. The
remainder after dividing by 11 gives a number from 0 to 10. Subtracting this from
11 gives a number from 1 to 11. That result is the check digit, except in the two
cases where it is 10 or 11. If 10 it is then changed to a 0, and if 11 then it is
changed to a 5. The check digit may be used to verify if a barcode scan is correct,
or if a manual entry of the barcode is correct.”
Application of the above procedure to the tracking number on the slip of paper shows that
the number on the slip of paper is not a valid tracking number as the final check digit
should be an 8 rather than the 0.
Can a probability be determined that a suspect would have a tracking number that
is off by one digit from the tracking number on the parcel?
In short, except for the trivial case, no. Computation of a formal probability depends on
the frequency of the event of interest divided by the set of possible events (assuming all
possible events are equally probable). The latter is impossible to determine except in the
trivial case.
For example, suppose that in a purse snatching, a customer lost a single Lotto 6/49 ticket.
The customer remembers what numbers were picked. Later a suspect is found to have a
Lotto 6/49 ticket in their possession with the same set of 6 numbers. What is the
probability that this could happen? [Ignoring information on place of purchase etc that
can be determined from the actual tickets.]
1
In the trivial case, if the suspect claims that he/she randomly picked the numbers on the
ticket in his/her possession, and it was only a coincidence that the six number matched,
then we can now compute a probability. The set of all possible events are all the possible
sets of 6 numbers that could be chosen in Lotto 6/49 (about 14,000,000). The probability
that the suspect would randomly pick a set of 6 numbers that would match the stolen
ticket is 1/14,000,000. While theoretically possible, it is not plausible that the suspect
would randomly pick six numbers that matched the stolen ticket.
However, suppose the suspect claims that the numbers represent the birthdays of favorite
movie stars and the numbers were not chosen at random. Now it is impossible to compute
a probability because it is impossible to know how many movie stars are known by the
suspect, and if the suspect knows the stars’ birthdays. About the only thing that can be
done in this case is to work backwards – are there indeed a set of movie stars whose
birthdays are known and match the numbers on the stolen ticket. If there is no set of six
movie stars with the corresponding birthdays then this would eliminate this rationale, but
no probability can be attached.
In the case of a parcel, if the number in possession of the suspect were generated at
random, then the probability for this trivial case can be determined. The set of possible
sets of 9 numbers that could be generated at random is one billion. There are 9 possible
tracking numbers that match the parcel in the first 8 of the 9 digits, but don’t match on
one digit. For example. there are 9 possibilities that look like 001 210 74x where x does
not match. There are also 9 possibilities where the tracking number could have
mismatched in the second to last digit etc. In total the probability is approximately
81/1,000,000,000 (81 in a billion) that a randomly generated set of 9 digits would be off
by 1 digit somewhere in the nine numbers compared to the parcel tracking number. So
under this scenario, while theoretically possible, it is not plausible that a random
generation of numbers would match in all but one digit.
Can the way the tracking numbers are generated be used to compute a probability?
The above analysis ignores the role of the check digit in parcel tracking numbers.
The set of numbers in possession of the suspect looks like, but are not a valid tracking
number. As noted in the report from Canada Post, the last digit in the number does not
occur at random but is computed using a formula. The tracking number should end in an
8 but the number on the suspect ends in a 0.
Single digit mistakes:
Suppose the suspect argues that the number in their possession is for a different parcel
but a mistake was made in one of the digits. For example, the suspect intended to write
down CP 101 201 740 SY but “accidentally” wrote down CP 001 201 740 SY. I
examined all possible tracking numbers that were off by one digit from the parcel
tracking number (e.g. 101 201 740, 201 210 740 etc) to determine if there are valid
tracking numbers for which an error could have lead to the close match. This list is found
in Table 1 at the end of this report. There were only 8 potential tracking numbers for
2
which a one digit error could lead to the close match. For example, the valid tracking
number 801 210 740 may have been written down as 001 210 740 leading to the close
match.
For each of these potential tracking numbers, there is a 1/10 chance that the digit that will
be transcribed in error would be chosen. What is the probability that the suspect would
indeed have one of these 8 potential valid tracking numbers? It should be straight forward
to have Canada Post examine this set of 8 potential tracking numbers to see if these have
been ever used or were in transit, and to where they where delivered. If none of the 8
potential tracking numbers that could lead to a “close match though a single digit
transcription” exist or be tracked by the suspect, then there would be no valid tracking
numbers that a suspect could have been tracking and this line of defense can be
eliminated.
Two digit transpositions:
A similar approach was used to see if there are any valid tracking numbers for which a
transposition of two digits would lead to the observed close match. I looked a all possible
tracking numbers which would result in the number in possession of the suspect if a two
digit transposition was “accidentally” made (Table 2). There were only two valid tracking
numbers (000 210 741 - transpose the 3rd and 9th digit; and 001 710 240 - transpose the
4th and 7th digit) for which a two digit transposition would result in a close match.
For each of these two potential tracking numbers, there is a 1/45 chance that those two
specific digits would be transposed to get the close match. Again, these numbers should
be checked with Canada Post and they have not been assigned or used, then this
eliminates this line of defense.
Three digit scrambling:
The tracking numbers are often written in groups of 3 (as done in this report). Perhaps a
transposition among a group of 3 digits occurred (e.g. the 201 in the middle really was
012). I examined all possible tracking numbers where a transposition among the digits in
one of the groups of 3 digits could have lead to the close match seen. There are no valid
tracking numbers for which this could have occurred indicating that this type of error
could not be used to “explain” why the number in possession of the suspect were a close
match to the parcel’s numbers.
Other errors:
There are many possible errors that could be examined in a similar fashion.
Please let me know if you require further information
Carl James Schwarz
3
Table 1
Check parcel codes off by 1 digit
Parcel
tracking
Obs code
Sum
Is
for
tracking
check Check code
digit digit valid?
1 101 210 740
110
5
2 201 210 740
118
3
3 301 210 740
126
6
4 401 210 740
134
9
5 501 210 740
142
1
6 601 210 740
150
4
7 701 210 740
158
7
8 801 210 740
166
0 yes
9 901 210 740
174
2
10 011 210 740
108
2
11 021 210 740
114
7
12 031 210 740
120
1
13 041 210 740
126
6
14 051 210 740
132
5
15 061 210 740
138
5
16 071 210 740
144
0 yes
17 081 210 740
150
4
18 091 210 740
156
9
19 000 210 740
98
1
20 002 210 740
106
4
21 003 210 740
110
5
22 004 210 740
114
7
23 005 210 740
118
3
24 006 210 740
122
0 yes
25 007 210 740
126
6
26 008 210 740
130
2
27 009 210 740
134
9
4
Table 1
Check parcel codes off by 1 digit
Parcel
tracking
Obs code
Sum
Is
for
tracking
check Check code
digit digit valid?
28 001 010 740
98
1
29 001 110 740
100
0 yes
30 001 310 740
104
6
31 001 410 740
106
4
32 001 510 740
108
2
33 001 610 740
110
5
34 001 710 740
112
9
35 001 810 740
114
7
36 001 910 740
116
5
37 001 200 740
99
5
38 001 220 740
105
5
39 001 230 740
108
2
40 001 240 740
111
0 yes
41 001 250 740
114
7
42 001 260 740
117
4
43 001 270 740
120
1
44 001 280 740
123
9
45 001 290 740
126
6
46 001 211 740
107
3
47 001 212 740
112
9
48 001 213 740
117
4
49 001 214 740
122
0 yes
50 001 215 740
127
5
51 001 216 740
132
5
52 001 217 740
137
6
53 001 218 740
142
1
54 001 219 740
147
7
55 001 210 040
39
5
5
Table 1
Check parcel codes off by 1 digit
Parcel
tracking
Obs code
Sum
Is
for
tracking
check Check code
digit digit valid?
56 001 210 140
48
7
57 001 210 240
57
9
58 001 210 340
66
5
59 001 210 440
75
2
60 001 210 540
84
4
61 001 210 640
93
6
62 001 210 840
111
0 yes
63 001 210 940
120
1
64 001 210 700
74
3
65 001 210 710
81
7
66 001 210 720
88
5
67 001 210 730
95
4
68 001 210 750
109
1
69 001 210 760
116
5
70 001 210 770
123
9
71 001 210 780
130
2
72 001 210 790
137
6
73 001 210 741
102
8
74 001 210 742
102
8
75 001 210 743
102
8
76 001 210 744
102
8
77 001 210 745
102
8
78 001 210 746
102
8
79 001 210 747
102
8
80 001 210 748
102
8 yes
81 001 210 749
102
8
6
Table 2
Check parcel codes with transpose of 2 digits
Parcel
tracking
Obs code
Sum
Is
for
tracking
Transpose Transpose
position
position check Check code
1
2 digit digit valid?
1 001 210 740
1
2
102
8
2 100 210 740
1
3
106
4
3 201 010 740
1
4
114
7
4 101 200 740
1
5
107
3
5 001 210 740
1
6
102
8
6 701 210 040
1
7
95
4
7 401 210 700
1
8
106
4
8 001 210 740
1
9
102
8
9 010 210 740
2
3
104
6
10 021 010 740
2
4
110
5
11 011 200 740
2
5
105
5
12 001 210 740
2
6
102
8
13 071 210 040
2
7
81
7
14 041 210 700
2
8
98
1
15 001 210 740
2
9
102
8
16 002 110 740
3
4
104
6
17 001 210 740
3
5
102
8
18 000 211 740
3
6
103
7
19 007 210 140
3
7
72
5
20 004 210 710
3
8
93
6
21 000 210 741
3
9
98
1 yes
22 001 120 740
4
5
103
7
23 001 012 740
4
6
108
2
24 001 710 240
4
7
67
0 yes
25 001 410 720
4
8
92
7
26 001 010 742
4
9
98
1
27 001 201 740
5
6
104
6
28 001 270 140
5
7
66
5
29 001 240 710
5
8
90
9
30 001 200 741
5
9
99
5
7
Table 2
Check parcel codes with transpose of 2 digits
Parcel
tracking
Obs code
Sum
Is
for
tracking
Transpose Transpose
position
position check Check code
1
2 digit digit valid?
31 001 217 040
6
7
74
3
32 001 214 700
6
8
94
5
33 001 210 740
6
9
102
8
34 001 210 470
7
8
96
3
35 001 210 047
7
9
39
5
36 001 210 704
8
9
74
3
8
Table 3
Check parcel codes with transpose among groups of 3 digits
Parcel
tracking
Obs code
Sum
Is
for
tracking
Start of
3-digit check Check code
group digit digit valid?
1 001 210 740
1
102
8
2 010 210 740
1
104
6
3 001 210 740
1
102
8
4 010 210 740
1
104
6
5 100 210 740
1
106
4
6 100 210 740
1
106
4
7 001 210 740
4
102
8
8 001 201 740
4
104
6
9 001 120 740
4
103
7
10 001 102 740
4
107
3
11 001 021 740
4
106
4
12 001 012 740
4
108
2
13 001 210 740
7
102
8
14 001 210 704
7
74
3
15 001 210 470
7
96
3
16 001 210 407
7
47
8
17 001 210 074
7
60
6
18 001 210 047
7
39
5
9