I don't follow. The example in the wikipedia uses the opposite error rates to what you have given: it falsely flags a sober driver in 5% of cases. And it works because the stipulated rate of drunkenness is much lower than 5% (0.1%) so the false positives overwhelm the true positives. In your setup the positive detection should be accurate at 95%, the negative error (a drunk driver passed as sober) - would be around 0.005%, where did you get 1.96%?

Furthermore, as Wikipedia article noted, this assumes total testing. I highly doubt cops are walking around and drug test everyone they see.