Does this type of classification exist?
Instead of formulating the problem with Venn diagrams you could also look at a simple two by two table. Usually the problem is formulated graphically in a different way (see picture below from Wikipedia page). If you are interested in just predicting the occurrence of the value 1 you are just focusing on the sensitivity (or true positive rate) of your classification algorithm.
This is rater straightforward within a ROC analysis framework, you could just select a minimun value for the classifier threshold. However, this comes at the cost of very low specificity. You should also consider the cost-benefit ratio of your sensitivity/specificity results. 

https://en.m.wikipedia.org/wiki/Sensitivity_and_specificity