A breathalyser is used to check whether a driver’s blood alcohol level is over t

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)= P(B)*P(A|B)+(1-P(B))*P(A|Bc)

=P(B)*0.9+(1-P(B))*0.01

P(A) =0.89*P(B)+0.01

therefore P(B|A) =P(B)*0.9/(0.89*P(B)+0.01)

b) here as P(B) =1-0.95 =0.05

hence error rate =0.05*0.9/(0.89*0.05+0.01)=0.8257

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