ATRACTOR

The check digit \(C\) of Visa cards is computed in the following way: \[\overline{2x_{1}}+x_{2}+\overline{2x_{3}}+x_{4}+\overline{2x_{5}}+x_{6}+\overline{2x_{7}}+x_{8}+\overline{2x_{9}}+x_{10}+\overline{2x_{11}}+x_{12}+\overline{2x_{13}}+x_{14}+\overline{2x_{15}}+C=0\,(\mbox{mod }10)\] where \(x_{1}\) is the first digit of the Visa number, \(x_{2}\) is the second digit, \(x_{3}\) is the third, etc. Furthermore: \[\overline{2x_{i}}=\left\{ \begin{array}{ll} \overline{2x_{i}} & \mbox{if }\overline{2x_{i}}< 10\\ \overline{2x_{i}}-9 & \mbox{if }\overline{2x_{i}}\geq10 \end{array}\right.\]

Note that we are using here Modular Arithmetic, not usual arithmetic.

What is the effectiveness rate of this system in errors detection?