Is the 10% given in the question the same as the 1/10 figure you are coming up with or a different one?

Same one.
Is that what you mean or something else?

Something else. For example, I want you to write down what it means for the false positive rate to be 0.005 in the form:
The probability of ________ is ____
or maybe
The probability of _______ given that __________ is _____
(or, preferably, the equivalent statement using P notation)