On each tile in a set of dominoes, each integer 0 through 18 is paired exactly once with each integer 0 through 18. One number of the pair is written on one half of the domino, and the other number is written on the other half. If one domino is selected at random from all the dominoes in this set, what is the probability that the same number will be on both halves? Express your answer as a common fraction in lowest terms.
