Question 1192508
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A breathalyzer test is used by police in an area to determine whether a driver has an excess of alcohol in their blood. 
The device is not totally reliable: 7% of drivers who have not consumed an excess of alcohol give a reading 
from the breathalyzer as being above the legal limit, while 10% of drivers who are above the legal limit 
will give a reading below that level. Suppose that in fact 14% of drivers are above the legal alcohol limit, 
and the police stop a driver at random. Give answers to the following to four decimal places.

(a) What is the probability that the driver is incorrectly classified as being over the limit?

(b) What is the probability that the driver is correctly classified as being over the limit?

(c) Find the probability that the driver gives a breathalyzer test reading that is over the limit.

(d) Find the probability that the driver is under the legal limit, given the breathalyzer reading is also below the limit.
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       In the post by @CPhill,  parts  (a),  (b)  and  (d)  are solved and answered  INCORRECTLY.

        I came to bring correct solutions to all parts.



<pre>
(a)  In (a), they want you determine the probability of two simultaneous events:
     the driver is under the limit, but the reading incorrectly classified him
     as being over the limit.

     The <U>ANSWER</U> to (a) is  (1-0.14)*0.07 = 0.0602 = 6.02%.



(b)  In (b), they want you determine the probability of two simultaneous events:
     the driver is over the limit, and the reading correctly classified him 
     as being over the limit.

     The <U>ANSWER</U> to (b) is  0.14*(1-0.1) = 0.126 = 12.6%.   



(c)  This probability is the sum of two probabilities of disjoint events:

         - the driver is under the limit (1-0.14), but the reading is mistakenly over the limit
               P1 = (1-0.14)*0.07;

         - the driver is over  the limit (0.14),   and the reading is correctly over the limit
               P2 = 0.14*0.9.

      Therefore, the probability in (c) is  P = P1 + P2= (1-0.14)*0.07 + 0.14*0.9 = 0.1862 = 18.62%.  <U>ANSWER</U> to (c)



(d)  In (d), the conditional probability is 

          P = {{{((1-0.14)*(1-0.07))/((1-0.14)*(1-0.07)+0.14*(1-0.9))}}} = {{{0.7998/0.8138}}} = 0.9830 = 98.30%.  <U>ANSWER</U> to (d)


     In this formula, the numerator is the probability of the event that the driver is under the legal limit 
     and the reading is under the legal limit.

     The denominator is the probability that the reading is below the legal limit.
     The structure of the denominator here is similar to expression in part (c).
</pre>

Solved.


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The goal of this problem is to teach a student to think logically.
Having and using common sense is enough to make every step.


The &nbsp;TWIN &nbsp;to this problem was solved at this forum many years ago under this link

https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1206991.html


This example on how incorrectly the problem was solved in the post by @CPhill shows
that the references to formal theorems do not save from making monstrous errors.


Common sense should work and really works much better in such simple problems.
And this is exactly my major goal in this post to develop the student's common sense.



The last answer in the post by @CPhill


* d) Probability of driver under the limit given breathalyser reading below limit: 0.0740


tells me that, in reality, nobody even read it and nobody even thought on/about it 
before submitting it to the forum.


It is below any common sense level.