Your right. Here's the flaw I think:

My first day there has a 1/40 chance of being one of the 9 days that she's there already.

But, if she's not there, my second day there has only a 1/120 chance of being one of her first days there of one of her three trips.

And so on, or my third and fourth days there. So, 3/120 + 1/120 + 1/120 + 1/120, or 6/120, or 19-1.