In case it's not clear (that wording doesn't really click for me, since there are no exponents involved), the way this information is encoded is in the shape of the distribution itself. If there have been very few observations, it will be very wide, but after many it will narrow to a tiny spike.

In this context, at least, the prior distribution encodes everything - there's no meaning to the idea that you're more or less confident in the prior, because the prior already represents your uncertainty about the outcomes. If you were 50/50 on this prior versus another one, then your actual prior would be the average of the two.

There is a subfield of statistics that deals with imprecise probabilities, but that's a whole other can of worms and doesn't really relate to this problem. That said, it's fascinating stuff, and very useful in some contexts (if you're uncertain about your priors, it can be useful to do sensitivity analyses to figure out exactly how the end result depends on your prior).