A breathalyser is used to check whether a driver’s blood alcohol level is over t
ID: 3061678 • Letter: A
Question
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?
Explanation / Answer
(a) Here P(A l B ) = 0.90 and P(A' l B') = 0.99
Pr(Rate of wrongful convictions) = Pr(Driver over the limit l Breathanalyser shows driver uner the limit) = Pr(A l B') = 1 - Pr(A'l B') = 1 - 0.99 = 0.01
(b) Here
if Pr(Drivers on a Saturday night are below the limit) = Pr(B') = 0.95
Error rate
Pr(Error rate) = Pr(B') * Pr(A l B') = 0.95 * 0.01 = 0.0095
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