The digit sum of a positive integer is the sum of all its digits. For example, the digit sum of the integer \(2345\) is \(2+3+4+5=14\).
Ayumi announced that her favourite number is a three-digit positive integer with a digit sum of \(9\). Vinnie then wrote down a three-digit positive integer with a digit sum of \(9\). What is the probability that the number Vinnie wrote was Ayumi’s favourite number?