What are the odds of the Digging Duo?
Solution 1:
As far as Serebii datamined, they found only the focused one had any chance of fetching the gold bottle caps.
On the image shown beyond, we can see a 0% to get the golden cap. This is most likely false since some people got it, yet it must be less than 1% hence it's very rare status.
This reddit link got some more data. But the most you can learn from it (except the datasheet link there) is that there is a special animation when you dig it.