Each of 9 girls participates in several (one or more) theater groups, so that there are no two identical groups. Each of them is randomly assigned a positive integer between 1 and 30 inclusive. We call a group \textit{small} if the sum of the numbers of its members does not exceed the sum of any other group. Prove that regardless of which girl participates in which group, the probability that after receiving the numbers there will be a unique small group is at least \( \frac{7}{10} \).