SOLUTION: (1 point) A breathalyser 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

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Question 1192508: (1 point) A breathalyser 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 breathalyser 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.
Part a)
What is the probability that the driver is incorrectly classified as being over the limit?

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

Part c)
Find the probability that the driver gives a breathalyser test reading that is over the limit.

Part d)
Find the probability that the driver is under the legal limit, given the breathalyser reading is also below the limit.

Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1987)   (Show Source): You can put this solution on YOUR website!
Certainly, let's break down this problem step-by-step.
**Let's define some events:**
* **A:** Driver is above the legal alcohol limit.
* **B:** Breathalyser reading is above the legal limit.
**Given Information:**
* P(A) = 0.14 (Probability of a driver being above the limit)
* P(B|A') = 0.07 (Probability of a false positive: breathalyser reading above limit when driver is not above limit)
* P(B'|A) = 0.10 (Probability of a false negative: breathalyser reading below limit when driver is above limit)
**Part a) Probability of Incorrectly Classified as Over the Limit (False Positive)**
* This is the probability of the breathalyser reading being above the limit given that the driver is *not* above the limit (A').
* P(B|A') = 0.07 (Given in the problem statement)
**Part b) Probability of Correctly Classified as Over the Limit (True Positive)**
* This is the probability of the breathalyser reading being above the limit given that the driver *is* above the limit (A).
* P(B|A) = 1 - P(B'|A) = 1 - 0.10 = 0.90
**Part c) Probability of a Breathalyser Reading Over the Limit**
* We need to use the Law of Total Probability:
P(B) = P(B|A) * P(A) + P(B|A') * P(A')
P(B) = (0.90 * 0.14) + (0.07 * 0.86)
P(B) = 0.126 + 0.0602
P(B) = 0.1862
**Part d) Probability of Driver Under the Limit Given Breathalyser Reading Below the Limit**
* This is the probability of not being above the limit (A') given that the breathalyser reading is below the limit (B').
* We'll use Bayes' Theorem:
P(A'|B') = [P(B'|A') * P(A')] / P(B')
* We need to find P(B'):
* P(B') = 1 - P(B) = 1 - 0.1862 = 0.8138
* P(A'|B') = [0.07 * 0.86] / 0.8138
* P(A'|B') = 0.0602 / 0.8138
* P(A'|B') = 0.0740
**In summary:**
* a) Probability of false positive: 0.0700
* b) Probability of true positive: 0.9000
* c) Probability of breathalyser reading over the limit: 0.1862
* d) Probability of driver under the limit given breathalyser reading below limit: 0.0740
Let me know if you have any other questions or need further clarification!
Answer by ikleyn(52946)   (Show Source): You can put this solution on YOUR website!
.
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.
~~~~~~~~~~~~~~~~


       In the post by @CPhill,  parts  (a),  (b)  and  (d)  are solved and answered  INCORRECTLY.
        I came to bring correct solutions to all parts. 


(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 ANSWER 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 ANSWER 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%.  ANSWER to (c)



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

          P =  =  = 0.9830 = 98.30%.  ANSWER 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).

Solved.

----------------------

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  TWIN  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.



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