When are you not allowed to drive a motorized bicycle on public roads?

You answered correctly by identifying that driving a motorized bicycle on public roads is prohibited in two specific situations.

1. **Lack of motor vehicle liability insurance**: In Germany, every motorized vehicle must have insurance to protect against damages that might occur during its operation. Without this insurance, you cannot legally drive on public roads.

2. **Unroadworthy condition of the motorized bicycle**: Vehicles must meet safety standards to ensure they are fit for use on public roads. If a motorized bicycle is not roadworthy, it poses risks not only to the driver but also to other road users.

It is important to recognize that while there are other considerations, such as not paying motor vehicle tax, the combination of the first two options directly relates to public safety and legal compliance. Understanding these rules helps to ensure safe and lawful driving practices.