SOLUTION: b. A manufacturer of a firm claims that 70% of its new hires turn out to be good workers and the rest are to be poor workers. The company advised their current workers to qualif

Question 1197314: b. A manufacturer of a firm claims that 70% of its new hires turn out to be good workers and the
rest are to be poor workers. The company advised their current workers to qualify a reasoning
test. Eighty percent of the good workers and forty percent of the poor workers pass the reasoning test. If the company makes the reasoning test be a part of its hiring procedure, what then is the
probability of the new hires that,
i.will turn out to be good workers? (Marks -4]
ii. will turn out to be poor workers? (Marks-4]
iii. Depict the turn out percentage of the workers on the Bayes' tree diagram. [Mark-1]
Answer by onyulee(41) (Show Source): You can put this solution on YOUR website!
**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.

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