Question 1176351
Let's solve this problem step-by-step.

**Given Information:**

* Total weeks: 52
* Overtime (O) weeks: 18
* Temporary help (T) weeks: 9
* Overtime and temporary help (O ∩ T) weeks: 5

**a) What is the probability P(O OR T)?**

We'll use the formula for the probability of the union of two events:

* P(O ∪ T) = P(O) + P(T) - P(O ∩ T)

1.  **Calculate P(O):**
    * P(O) = Number of overtime weeks / Total weeks
    * P(O) = 18 / 52

2.  **Calculate P(T):**
    * P(T) = Number of temporary help weeks / Total weeks
    * P(T) = 9 / 52

3.  **Calculate P(O ∩ T):**
    * P(O ∩ T) = Number of weeks with both / Total weeks
    * P(O ∩ T) = 5 / 52

4.  **Calculate P(O ∪ T):**
    * P(O ∪ T) = (18 / 52) + (9 / 52) - (5 / 52)
    * P(O ∪ T) = (18 + 9 - 5) / 52
    * P(O ∪ T) = 22 / 52
    * P(O ∪ T) = 11 / 26
    * P(O ∪ T) ≈ 0.4230769

5.  **Round to the nearest hundredth:**
    * P(O ∪ T) ≈ 0.42

**b) Are O and T independent?**

Two events are independent if P(O ∩ T) = P(O) * P(T).

1.  **Calculate P(O) * P(T):**
    * P(O) * P(T) = (18 / 52) * (9 / 52)
    * P(O) * P(T) = 162 / 2704
    * P(O) * P(T) ≈ 0.0599

2.  **Compare P(O ∩ T) and P(O) * P(T):**
    * P(O ∩ T) = 5 / 52 ≈ 0.0961
    * P(O) * P(T) ≈ 0.0599

3.  **Conclusion:**
    * Since P(O ∩ T) ≠ P(O) * P(T), the events O and T are **not independent**.

**c) Are O and T mutually exclusive?**

Two events are mutually exclusive if P(O ∩ T) = 0.

1.  **Check P(O ∩ T):**
    * P(O ∩ T) = 5 / 52

2.  **Conclusion:**
    * Since P(O ∩ T) ≠ 0, the events O and T are **not mutually exclusive**.

**Answers:**

* **a) P(O OR T) ≈ 0.42**
* **b) No, O and T are not independent.**
* **c) No, O and T are not mutually exclusive.**