Crowdsourced annotations of data play a substantial role in the development of Artificial Intelligence (AI). It is broadly recognised that annotations of text data can contain annotator bias, where systematic disagreement in annotations can be traced back to differences in the annotators' backgrounds. Being unaware of such annotator bias can lead to representational bias against minority group perspectives and therefore several methods have been proposed for recognising bias or preserving perspectives. These methods typically require either a substantial number of annotators or annotations per data instance. In this study, we propose a simple method for handling bias in annotations without requirements on the number of annotators or instances. Instead, we ask annotators about their beliefs of other annotators' judgements of an instance, under the hypothesis that these beliefs may provide more representative and less biased labels than judgements. The method was examined in two controlled, survey-based experiments involving Democrats and Republicans (n=1,590) asked to judge statements as arguments and then report beliefs about others' judgements. The results indicate that bias, defined as systematic differences between the two groups of annotators, is consistently reduced when asking for beliefs instead of judgements. Our proposed method therefore has the potential to reduce the risk of annotator bias, thereby improving the generalisability of AI systems and preventing harm to unrepresented socio-demographic groups, and we highlight the need for further studies of this potential in other tasks and downstream applications.