Question 1192522
Certainly, let's break down the problem and find the probabilities.

**a) Probability of a False Positive**

* **False Positive:** This occurs when an individual *does not* have diabetes but is *incorrectly* diagnosed as having it.

* **Calculation:**
    * Probability of not having diabetes: 1 - 0.0837 = 0.9163
    * Probability of false positive diagnosis given no diabetes: 0.03

    * Probability of false positive = 0.9163 * 0.03 = **0.0275**

**b) Probability of Being Diagnosed as Not Having Diabetes**

* **This involves two scenarios:**
    * 1. **True Negative:** Individual does not have diabetes and is correctly diagnosed as not having it. 
        * Probability: 0.9163 (probability of not having diabetes) * 0.97 (probability of correct diagnosis given no diabetes) = 0.8895

    * 2. **False Negative:** Individual has diabetes but is incorrectly diagnosed as not having it.
        * Probability: 0.0837 (probability of having diabetes) * 0.02 (probability of incorrect diagnosis given diabetes) = 0.0017

    * **Total Probability of Being Diagnosed as Not Having Diabetes:** 0.8895 + 0.0017 = **0.8912**

**c) Probability of False Negative**

* **False Negative:** As defined above, this is the probability of having diabetes but being incorrectly diagnosed as not having it.

* **Calculation:** 
    * From part (b), we calculated the probability of a false negative as 0.0017.

    * To find the conditional probability of actually having diabetes given that the diagnosis is negative:
        * Probability of false negative / Probability of being diagnosed as not having diabetes 
        * = 0.0017 / 0.8912 
        * = **0.0019**

**In summary:**

* a) Probability of false positive: 0.0275
* b) Probability of being diagnosed as not having diabetes: 0.8912
* c) Probability of false negative: 0.0019

**Key takeaway:** While the diagnostic test is generally accurate, there's still a small but significant risk of false positives and false negatives.