A breathalyser is used to check whether a driver’s blood alcohol level is over the legal limit. We define these events:

A : breathalyser shows driver is over the limit;
B : the driver’s blood alcohol really is over the limit.

(a) Suppose

P ( A | B ) = 0.9, P ( A ̄| B ̄) = 0.99

so that the breathalyser is reasonably reliable. A driver is stopped at random and breathalysed. If a conviction is made on the basis of the breathalyser alone, deter- mine the error rate (rate of wrongful convictions)

̄ P ( B | A ) = P (driver is under the limit breathalyser shows driver over the limit).

Hints: For a given two events C and D , it holds true that P ( C | D ) + P ( C | D ) = 1, where C denotes the complement of the event C .

(b) Suppose 95% of the drivers on a Saturday night are below the limit. What is the error rate?