1. **Problem Statement:** We have a table showing the number of employees classified by age groups and departments. We need to calculate various probabilities based on this data. 2. **Data from the table:** - Age <30: Production = 50, Sales = 2, Office = 50 - Age 30-45: Production = 70, Sales = 26, Office = 48 - Age >45: Production = 36, Sales = 4, Office = 14 3. **Total employees:** $$\text{Total} = (50+2+50) + (70+26+48) + (36+4+14) = 102 + 144 + 54 = 300$$ 4. **(a) Probability employee is under 30 years old:** Number under 30 = $50 + 2 + 50 = 102$ $$P(\text{under 30}) = \frac{102}{300} = \frac{17}{50} = 0.34$$ 5. **(b) Probability employee works in sales department:** Number in sales = $2 + 26 + 4 = 32$ $$P(\text{sales}) = \frac{32}{300} = \frac{8}{75} \approx 0.1067$$ 6. **(c) Probability employee is an office worker and above 45 years:** Number office and >45 = 14 $$P(\text{office and >45}) = \frac{14}{300} = \frac{7}{150} \approx 0.0467$$ 7. **(d) Probability employee is over 45 years given that he is an office worker:** Total office workers = $50 + 48 + 14 = 112$ Number office and >45 = 14 Conditional probability: $$P(>45 | \text{office}) = \frac{14}{112} = \frac{1}{8} = 0.125$$ 8. **(e) Probability employee is a production officer given that he is between 30 and 45 years:** Total employees aged 30-45 = $70 + 26 + 48 = 144$ Number production and 30-45 = 70 Conditional probability: $$P(\text{production} | 30 \leq \text{age} \leq 45) = \frac{70}{144} = \frac{35}{72} \approx 0.4861$$