Why Are Credit Card Numbers So Long?

A single digit can have one of ten possible values: 0, 1, 2,..., 9. For a two-digit number, such as 47, there are ten possibilities for the first digit and, for each of these values, ten possibilities for the second digit, giving 10² = 100 possible values in all. (Alternatively, we can reason directly that there are 100 possibilities, ranging from 00 to 99.) A three-digit number has 10³ = 1,000 possible values. A five-digit number has 10⁵ = 100,000 possible values.

The number of possible ten-digit numbers is 10¹⁰ = 10 billion, enough to give a unique 10-digit identification number to every person who is now alive or will be born in the next 50 years. Yet, Visa credit cards have a minimum of 13 digits, American Express cards have 15 digits, and MasterCards have 20 digits. Why do these credit cards have many, many more digits than they will ever need to keep track of their cardholders? Don’t these extra, superfluous digits make it more likely that a clerk filling out a sales ticket, taking an order over the phone, or transferring numbers from sales receipts to a computer will make a mistake, causing the charge to be billed to someone else’s number?

In fact, the additional digits are a security device, intended to make erroneous billings less likely, by making it extremely improbable that an incorrectly entered number will belong to someone. MasterCard’s 20-digit system creates 10²⁰ possible numbers, of which approximately 75 million are randomly assigned to its active cardholders. If a clerk makes a mistake, the probability that this randomly mistaken number will belong to a cardholder is 75,000,000 / 10²⁰ which is less than 1 in a trillion. In virtually every case, a computer will reject an incorrectly entered number and ask the clerk to try again. Even with Visa’s relatively modest 13-digit system, the probability that a randomly selected number will belong to a cardholder is less than 1 in 100,000. Credit card companies use long numbers to put probabilities on their side, and on ours.

Smith, Gary. Statistical Reasoning. 3rd edition. United States: McGraw-Hill, Inc., 1994.