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\n" ); document.write( "Question:
\n" ); document.write( "If you were to make up a random 13-digit number, what is the probability that it could pass as a legitimate ISBN based on the check digit?
\n" ); document.write( "
\n" ); document.write( "Solution:
\n" ); document.write( "First we have to know how the check digit works for an ISBN number.
\n" ); document.write( "Say the ISBN13 number is
\n" ); document.write( "9780486462370
\n" ); document.write( "The last digit is the check digit, and will not enter into the following calculations.
\n" ); document.write( "Step 1:
\n" ); document.write( "calculate the sum of all the even and odd position digits. Do not add the check digit, which is counted as the first odd position.
\n" ); document.write( "sum of odd positions: 3+6+6+4+8+9=36=S1
\n" ); document.write( "sum of even positions: 7+2+4+8+0+7=28=S2
\n" ); document.write( "Step 2:
\n" ); document.write( "calculate the checksum = S1+3S2=36+3*28=120
\n" ); document.write( "Step 3:
\n" ); document.write( "Find the remainder of the checksum divided by 10.
\n" ); document.write( "120/10=12 remainder 0.
\n" ); document.write( "Step 4:
\n" ); document.write( "Subtract the remainder by 10 to find the check digit.
\n" ); document.write( "If the check digit is 10, take it as zero.
\n" ); document.write( "10-0=10 therefore the check digit is zero, which is exactly what's in the last digit.
\n" ); document.write( "
\n" ); document.write( "To proceed to calculate the probability, we first examine the calculation of the check digit.
\n" ); document.write( "It relates to the sum of the random odd digits and 3 times the sum of even digits. Knowing that all digits are random, we can almost say that the check digit is also random, i.e. with a probability of 1/10.
\n" ); document.write( "However, we have to confirm if the multiplication by 3 (of the even digits) yield a digit evenly distribution in the [0-9] range.
\n" ); document.write( "Check the final digit of the multiplication by three:
\n" ); document.write( "0 0
\n" ); document.write( "1 3
\n" ); document.write( "2 6
\n" ); document.write( "3 9
\n" ); document.write( "4 2
\n" ); document.write( "5 5
\n" ); document.write( "6 8
\n" ); document.write( "7 1
\n" ); document.write( "8 4
\n" ); document.write( "9 7
\n" ); document.write( "Thus, multiplication by three merely changes the digit in an evenly distributed over the range of [0-9], so therefore remains random.
\n" ); document.write( "
\n" ); document.write( "Since the check digit is randomly distributed over the range [0-9], we can say that a randomly created 13-digit number will qualify as an ISBN13 number with a probability of 1/10. \n" ); document.write( "