"False positives are rare" does not imply that a positive result it is probably true. True positives might be even rarer than false positives. The problem is f

"False positives are rare" does not imply that a positive result it is probably true. True positives might be even rarer than false positives. The problem is false positive rates are usually expressed in terms of "what proportion of negative cases will be reported as positive?" which can't answer the question "what proportion of positive results will be wrong?". Answering the latter also depends on how many true positives there are, which may vary, or even be completely unknown. The false negative rate also needs to be factored in.