Question

SHOW ALL WORK. Police often set up sobriety checkpoints — roadblocks where
drivers are asked a few brief questions to allow the officer to judge whether or not the
person may have been drinking. If the officer does not suspect a problem, drivers are
released to go on their way. Otherwise, drivers are detained for a Breathalyzer test that
will determine whether or not they will be arrested. The police say that based on the
brief initial stop, trained officers can make the right decision 80% of the time. Suppose
the police operate a sobriety checkpoint after 9:00 p.m. on a Saturday night, a time
when national traffic safety experts suspect that about 12% of drivers have been
drinking.
a) You are stopped at the checkpoint and, of course, have not been drinking. What’s
the probability that you are detained for further testing?
b) What’s the probability that any given driver will be detained?
c) What’s the probability that a driver who is detained has actually been drinking?
d) What’s the probability that a driver who was released had actually been drinking?
For ages 30, 40, and 49, determine the probability of there actually existing a health issue if the test produces a positive result. Hint: this question is asking three separate questions, one for each age group.