Probability Level 5

Bob wants to keep a good-streak on Brilliant, so he logs in each day to Brilliant in the month of June. But he doesn't have much time, so he selects the first problem he sees, answers it randomly and logs out, despite whether it is correct or incorrect.

Assume that Bob answers all problems with $\frac{7}{13}$ probability of being correct. He gets only 10 problems correct, surprisingly in a row, out of the 30 he solves. If the probability that happens is $\frac{p}{q}$, where $p$ and $q$ are coprime positive integers, find the last $3$ digits of $p+q$.

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