Question 1178959: Ownership status Size
Small Large
Private 68 45
Public 33 76
(a) Suppose a company is selected at random. Then compute the probability that
I. The company is private or it is large.
II. The company is small and publicly owned.
(b) Are being publicly owned and being a large company independent events?
(c) What is the value of P(small and large? Explain your answer.
(d) If two small firms at randomly selected what is the probability that both are private?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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
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