# Precision and recall clarification

I'm reading the book Fundamentals of Machine Learning for Predictive Data Analytics by Kelleher, et al. I've come across something that I think as an error but I want to check to be sure. When explaining precision and recall the authors write:

Email classification is a good application scenario in which the different information provided by precision and recall is useful. The precision value tells us how likely it is that a genuine ham email could be marked as spam and, presumably, deleted: 25% (1 − precision). Recall, on the other hand, tells us how likely it is that a spam email will be missed by the system and end up in our inbox: 33.333% (1 − recall).

Precision is defined as: $$TP \over {TP + FP}$$. Thus: $$1 - precision = 1 - {TP \over TP+FP} = {FP \over TP + FP} = P(\textrm{prediction incorrect}|\textrm{prediction positive})$$ So this should give us the probability that an email marked as ham (positive prediction) is actually spam. So precision and recall in the quote above should be switched?

• It's a good observation that ham is not necessarily positive. But in this case: P(predict spam|actual ham)=${FS \over TH+FS}$. 1 - precision with spam=positive gives $FS \over TS + FS$ – snowape Jun 16 '20 at 8:30