Question 1192508
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!