Or, for the approximate answer, just compare the base rate with the false positive rate (multiplying by .9something has small impacts that mostly cancel out). About 1% of people test positive due to having the disease (a bit less, actually), about 5% of people test positive because of an inaccurate test (a bit less, actually), so a person with a positive test has about a 1 in 6 chance of having the disease.
Or, for the approximate answer, just compare the base rate with the false positive rate (multiplying by .9something has small impacts that mostly cancel out). About 1% of people test positive due to having the disease (a bit less, actually), about 5% of people test positive because of an inaccurate test (a bit less, actually), so a person with a positive test has about a 1 in 6 chance of having the disease.