The size of an individual cell type, such as a red blood cell, does not vary much among humans. We use this knowledge as a prior for classifying and detecting cells in images with only a few ground truth bounding box annotations, while most of the cells are annotated with points. This setting leads to weakly semi-supervised learning. We propose replacing points with either stochastic (ST) boxes or bounding box predictions during the training process. The proposed "mean-IOU" ST box maximizes the overlap with all the boxes belonging to the sample space with a class-specific approximated prior probability distribution of bounding boxes. Our method trains with both box- and point-labelled images in conjunction, unlike the existing methods, which train first with box- and then point-labelled images. In the most challenging setting, when only 5% images are box-labelled, quantitative experiments on a urine dataset show that our one-stage method outperforms two-stage methods by 5.56 mAP. Furthermore, we suggest an approach that partially answers "how many box-labelled annotations are necessary?" before training a machine learning model.