Interior insulation of a non-insulated wall = mold risk

Good day dear experts,

we do not want to build a house, but we hope that maybe someone can still help us with a "building question".

We are considering renting a ground floor apartment in an old farmhouse. All rooms are insulated from the inside, only the living room (one wall) was not insulated.
Now we fear that water could collect on the cold wall and mold could form.

Is our fear justified? Or are we exaggerating?

Although you actually addressed experts, I am unfortunately not one either, but purely physically it can of course be the case that moisture from the room air condenses at the coldest spot. However, this depends on many factors that we cannot assess here, not even as experts. One thing is relatively certain: if there should be a problem with moisture, it will first show in the corners on the cold wall, and there the risk of mold may possibly exist.

For example, how high is the humidity? It would tend to be worse in the bathroom or kitchen. I have also seen mold in the bedroom, where less heating is used and at night every person releases moisture into the room air. Cold air can hold less moisture, so it condenses more easily.

How much moisture is added to the room by the presence of people, aquarium, plants, etc.?

How well is the room ventilated? Can moisture be carried outside at all, or is additional moisture or cold even brought in under certain circumstances by continuous ventilation?

How much is the room heated? How high is the temperature difference between uninsulated and insulated walls?

What material is the wall surface made of? At least the risk of mold also depends on that.

All questions that one could consider when wanting to give a definitive answer. If in doubt, simply ask the previous tenants about their experiences, or hire an expert to take a look at the premises.

That depends on many factors. However, it is much more likely that it is molding behind the insulation.