Precision uses the predicted-positive set TP+FP; recall uses the actual-positive set TP+FN. Accuracy includes TN and all cases. F1=2TP/(2TP+FP+FN); where precision P and recall R permit, it is 2PR/(P+R), a harmonic rather than arithmetic balance. F1 does not include TN and does not encode every task’s error costs. High recall can coexist with many false positives.

Zero denominators remain undefined here. No predicted positives makes precision 0/0; no actual positives makes recall 0/0. F1 is0 when TP 0 and FP+FN is positive, but undefined when all cases are TN. Library zero_division choices may replace undefined values with 0,1 or a missing value; this page displays the raw undefined ratio explicitly. These are unweighted binary positive-class metrics, with no multiclass averaging.

Case IDs are fixed synthetic examples: Rare positives has actual-positive Cases 1/2, Balanced has 1..6, No positives has none. Scenarios change the labelled toy dataset and reset its predictions, not a real population. Editing predictions does not demonstrate training, a causal intervention, calibrated probabilities or expected future accuracy. Choosing the right metric requires task priorities, representative data and acceptable error costs beyond these counts.