You are a State Trooper on a roadblock. Since it is New Year’s Eve, you know from prior experience that approximately 5% of the drivers on the road have blood alcohol content over the legal limit of 0.08%. Your portable breathalyzer gives a PASS / FAIL reading. From prior experience and manufacturer’s data, you know that the breathalyzer has a “hit rate” of 90% and a “false positive rate” of 20%. That means that if the breathalyzer says FAIL, there is a 90% chance of it being correct if the subject is legally DUI and a 20% chance of a FAIL reading even if the subject is not legally intoxicated. You stop a driver at random and test him. He passes, so you let him go. What is the probability that the driver was legally DUI?

Something sounds weird. 90% + 20% = 110%

I do not agree that the false positive rate (on non-DUI) and the correct positive rate (on true DUI)have to add up to 100%

He could either be truly DUI and not caught or non-DUI and correctly passed. They are asking about the first possibility.

You know (because it says so) that 5% or the people driving are DUI at that time.

The probability that he was DUI and passed anyway was 0.05*0.1 = 0.005

To calculate the probability that the driver was legally DUI given that he passed the breathalyzer test, we can use Bayes' theorem. Bayes' theorem allows us to update the probability based on new evidence.

Let's define the events:
A = Driver is legally DUI
B = Driver passes the breathalyzer test

We need to calculate P(A|B), which represents the probability of the driver being legally DUI given that he passes the breathalyzer test.

According to Bayes' theorem, we can express P(A|B) as:

P(A|B) = (P(B|A) * P(A)) / P(B)

Where:
P(B|A) = Probability of passing the breathalyzer test given that the driver is legally DUI
P(A) = Probability of the driver being legally DUI (0.05 or 5%)
P(B) = Probability of passing the breathalyzer test (overall probability)

To calculate P(B), we need to consider both scenarios: the driver being legally DUI and the driver not being legally DUI.

P(B) = P(B|A) * P(A) + P(B|A') * P(A')

Where:
P(B|A) = Probability of passing the breathalyzer test given that the driver is legally DUI (1 - 0.9 or 0.1)
P(A') = Probability of the driver not being legally DUI (1 - P(A) or 0.95)
P(B|A') = Probability of passing the breathalyzer test given that the driver is not legally DUI (1 - 0.2 or 0.8)

Plugging in the values:

P(B) = (0.1 * 0.05) + (0.8 * 0.95)
= 0.005 + 0.76
= 0.765

Now we have all the values needed to calculate P(A|B):

P(A|B) = (0.1 * 0.05) / 0.765
= 0.005 / 0.765
≈ 0.0065

Therefore, the probability that the driver was legally DUI given that he passed the breathalyzer test is approximately 0.0065 or 0.65%.