Please note, your implied definition of ‘random’ above is, well… strictly not random. Well, at least not in terms of determining within each survey whether or not to show this ‘random’ question. As defined - “of 10 households visited, 5 should show the question” - would require randomly picking a subset of 5 of the 10 a priori, and somehow communicating this pre-selected list to the device when filling in the 10 (and no more or no less) surveys. Picking a random number of 0 or 1 within each form will, on average, show the question 50% of the time, but there’s no guarantee that it will only show up on exactly 5 of the 10 surveys, or even show up on any of them at all!
Strictly speaking there is nothing (statistically) preventing exactly this from happening (without either a priori determining the 5 of 10, or somehow checking all previous instances - which then makes it no longer ‘random’).