In your terminology, they are detection rates for *different types of
all members which are the same.
You may say, they both* answer questions of the type "What fraction of cases do we correctly recognize?". But the difference is in the denominator of the fraction (the population this figure of merit refers to):
- Sensitivity answers the question: Of all patients that truly have the disease, what fraction do we correctly recognize as diseased.
You may say, sensitivity is the detection rate of diseased patients.
- Specificitiy answers the question: Of all patients that truly do not have the disease, what fraction do we correctly recognize as not diseased.
You may say, specificity is the detection rate of healthy people.
So they look at disjunct (sub)populations, and their denominators do have nothing in common. This makes them in first approximation independent of each other:
- A good test has both high sensitivity and high specificity (A in the diagram below), and
a test that has both low sensitivity and low specificity is a bad test (C).
It is often possible to some extent to trade-off some sensitivity for increased specificity and vice versa (this is what the 2nd video is about). For test B, this trade-off is along the dotted line. The dotted line will always go from high specificity and low sensitivity to high sensitivity and low specificity, never from "bad" to "very good".
It is still important as in some situations sensitivity is more important and in others specificity is more important (sometimes false negatives are worse than false positives, sometimes it is the other way round)-
| . . . * very good test A
| . . .
| * test B
| * bad test C .
* there are more such "What fraction do we get correct?"-type figures of merit that use yet different denominators, e.g. the predictive values in the 1st video.