I also like your rigorous and formal set theoretic approach, but I prefer in this case to be more informal.
As you mentioned, we can assume the events are independent, and that there was randomness to the event of overhearing students, e.g., the listener wasn't hanging out in a place exclusively frequented by 1030 students.
Perhaps it's due to my having a small brain, but it seemed easiest for me to reduce the problem to a completely different but equivalent problem. Say we have 80 geese. We break them into two groups, one with 60 geese and one with 20 geese. Of the 60 geese, we mark 10% of them, that is, 6 of them. Of the 20 geese, we mark 15% of them, that is, 3 of them. We then release them into the wild. Later we find a goose that's marked. What's the probability that the goose was originally in the group with 60 geese?
It should be immediately clear that the probability is 6/9 = 2/3.
Edit: Actually I'm not as confident now as a moment ago that the problem is equivalent. I'll leave it up for consideration but if it's in error I'll try to correct it.