Yesterday we looked at #1 graphically. We constructed a formula for the fraction of those people testing positive who do not use drugs in terms of the fraction of the population who use drugs and in terms of the fraction of the time the test produces correct results.

x: the fractionof the time the test is accuarate

y: the fraction of the population who use drugs

z: the desired upper limit for the fraction of those people testing positive who do not use drugs

We established yesterday that . Here I will solve this equation for x, so that it is expressed in terms of y and z.

We know that the fraction of the population who use drugs is 0.05. We also know that we want no more than 0.01 of the people who test positive to be non-drug users. When we substitute 0.01 for z and 0.05 for y in the above equation, we get , which, when calculated, produces x = 0.999469.

That is, the test must be at least 99.9469% accurate when testing at random in a population that is 5% drug users before the disciplinary board can be confident that, in the long run, it will wrongly discipline no more than 1% of the people coming before it.