Question 1178959
Let's break down this problem step-by-step:

**1. Create a Total Table:**

| Ownership Status | Small | Large | Total |
|------------------|-------|-------|-------|
| Private          | 68    | 45    | 113   |
| Public           | 33    | 76    | 109   |
| Total            | 101   | 121   | 222   |

**a) Probability Calculations:**

* **I. The company is private or it is large.**
    * P(Private) = 113/222
    * P(Large) = 121/222
    * P(Private AND Large) = 45/222
    * P(Private OR Large) = P(Private) + P(Large) - P(Private AND Large)
    * P(Private OR Large) = (113/222) + (121/222) - (45/222) = 189/222 = 63/74 ≈ 0.8514

* **II. The company is small and publicly owned.**
    * P(Small AND Public) = 33/222 = 11/74 ≈ 0.1486

**b) Independence of Public Ownership and Large Size:**

* To check for independence, we need to see if P(Public AND Large) = P(Public) * P(Large).
* P(Public AND Large) = 76/222 = 38/111 ≈ 0.3423
* P(Public) = 109/222 ≈ 0.4910
* P(Large) = 121/222 ≈ 0.5450
* P(Public) * P(Large) = (109/222) * (121/222) ≈ 0.2677
* Since P(Public AND Large) ≠ P(Public) * P(Large), the events are **not independent**.

**c) P(Small and Large)?**

* P(Small AND Large) = 0/222 = 0
* This is because a company cannot be both "small" and "large" simultaneously. These are mutually exclusive categories within the provided data.

**d) Probability of Two Small Firms Being Private:**

* P(Small AND Private) = 68/101
* We need to find the probability of selecting two small firms that are both private.
* P(1st small firm is private) = 68/101
* P(2nd small firm is private, given the 1st was private) = 67/100 (since we assume selections are without replacement)
* P(Both are private) = (68/101) * (67/100) = 4556/10100 ≈ 0.4511

**Answers:**

* **(a) I.** 189/222 or approximately 0.8514
* **(a) II.** 33/222 or approximately 0.1486
* **(b)** No, they are not independent.
* **(c)** 0, because a firm can't be both small and large.
* **(d)** 4556/10100 or approximately 0.4511