Question 1197314
**i. Probability of a new hire being a good worker given they pass the test:**

* **Let:**
    * G: Event that a new hire is a good worker
    * P: Event that a new hire passes the reasoning test 

* **Given:**
    * P(G) = 0.70 (Probability of a good worker)
    * P(P|G) = 0.80 (Probability of passing the test given they are a good worker)
    * P(P|G') = 0.40 (Probability of passing the test given they are a poor worker)
    * P(G') = 1 - P(G) = 0.30 (Probability of a poor worker)

* **Use Bayes' Theorem:**

   P(G|P) = [P(P|G) * P(G)] / [P(P|G) * P(G) + P(P|G') * P(G')]

   P(G|P) = [0.80 * 0.70] / [(0.80 * 0.70) + (0.40 * 0.30)] 
   P(G|P) = 0.56 / (0.56 + 0.12)
   P(G|P) = 0.56 / 0.68 
   P(G|P) ≈ 0.8235 

**Therefore, the probability of a new hire being a good worker given they pass the reasoning test is approximately 0.8235 or 82.35%.**

**ii. Probability of a new hire being a poor worker given they pass the test:**

* **Use the complement rule:**
   * P(G'|P) = 1 - P(G|P) 
   * P(G'|P) = 1 - 0.8235 
   * P(G'|P) = 0.1765

**Therefore, the probability of a new hire being a poor worker given they pass the reasoning test is approximately 0.1765 or 17.65%.**

**iii. Bayes' Tree Diagram**

* **Root Node:** "New Hire"
* **Branch 1:** "Good Worker" (0.70) 
    * Sub-branch: "Passes Test" (0.80) 
    * Sub-branch: "Fails Test" (0.20)
* **Branch 2:** "Poor Worker" (0.30)
    * Sub-branch: "Passes Test" (0.40) 
    * Sub-branch: "Fails Test" (0.60)

**Note:** The probabilities on the branches of the tree diagram represent the conditional probabilities given the preceding event.

This analysis shows that implementing the reasoning test significantly increases the likelihood of hiring good workers.