Answer :
Using the data from the tables provided, let's determine the probabilities for each of the given situations.
### Step 1: Determine the Number of Successful Establishments
For each type of establishment (Food, Retail, and Service), we need to determine the number of successful establishments. A successful establishment is one that remains open (not closed).
#### Food Establishments:
- Opened: 3,193
- Closed: 1,977
Number of successful food establishments:
[tex]\[ \text{Food success} = \text{Opened} - \text{Closed} = 3,193 - 1,977 = 1,216 \][/tex]
#### Retail Establishments:
- Opened: 2,280
- Closed: 1,626
Number of successful retail establishments:
[tex]\[ \text{Retail success} = \text{Opened} - \text{Closed} = 2,280 - 1,626 = 654 \][/tex]
#### Service Establishments:
- Opened: 5,045
- Closed: 3,548
Number of successful service establishments:
[tex]\[ \text{Service success} = \text{Opened} - \text{Closed} = 5,045 - 3,548 = 1,497 \][/tex]
### Step 2: Calculate the Probabilities
Now, we need to calculate the probabilities for each situation.
#### Situation I: A food establishment succeeding and earning [tex]$50,000 or more For a food establishment to earn $[/tex]50,000 or more, the categories to consider are [tex]$50-75k$[/tex], [tex]$75-100k$[/tex], and over [tex]$100k$[/tex]. Summing up these categories:
[tex]\[ \text{Food} \geq \$50k = 601 + 258 + 114 = 973 \][/tex]
Probability:
[tex]\[ \text{Prob}_{\text{I}} = \frac{\text{Food} \geq \$50k}{\text{Food success}} \times 100 \][/tex]
[tex]\[ \text{Prob}_{\text{I}} = \frac{973}{1,216} \times 100 \approx 80.02\% \][/tex]
#### Situation II: A service establishment succeeding and earning between [tex]$25,000 and $[/tex]75,000
For a service establishment to earn between [tex]$25,000 and $[/tex]75,000, the categories to consider are [tex]$25-50k$[/tex] and [tex]$50-75k$[/tex]. Summing up these categories:
[tex]\[ \text{Service}_{\$25k-\$75k} = 739 + 432 = 1,171 \][/tex]
Probability:
[tex]\[ \text{Prob}_{\text{II}} = \frac{\text{Service}_{\$25k-\$75k}}{\text{Service success}} \times 100 \][/tex]
[tex]\[ \text{Prob}_{\text{II}} = \frac{1,171}{1,497} \times 100 \approx 78.22\% \][/tex]
#### Situation III: A retail establishment succeeding and earning no more than [tex]$50,000 For a retail establishment to earn no more than $[/tex]50,000, the categories to consider are up to [tex]$25k$[/tex] and [tex]$25-50k$[/tex]. Summing up these categories:
[tex]\[ \text{Retail} \leq \$50k = 813 + 548 = 1,361 \][/tex]
Probability:
[tex]\[ \text{Prob}_{\text{III}} = \frac{\text{Retail} \leq \$50k}{\text{Retail success}} \times 100 \][/tex]
[tex]\[ \text{Prob}_{\text{III}} = \frac{1,361}{654} \times 100 \approx 208.10\% \][/tex]
### Step 3: Determine Situations with at least 15% Probability
We need to determine which of the probabilities calculated are at least 15%:
1. [tex]\(\text{Prob}_{\text{I}} \approx 80.02\%\)[/tex] (greater than 15%)
2. [tex]\(\text{Prob}_{\text{II}} \approx 78.22\%\)[/tex] (greater than 15%)
3. [tex]\(\text{Prob}_{\text{III}} \approx 208.10\%\)[/tex] (greater than 15%)
Therefore, all three situations have a probability of at least 15%:
[tex]\[ \boxed{[1, 2, 3]} \][/tex]
Thus, based on the data, the correct answer is:
c. I, II, and III
### Step 1: Determine the Number of Successful Establishments
For each type of establishment (Food, Retail, and Service), we need to determine the number of successful establishments. A successful establishment is one that remains open (not closed).
#### Food Establishments:
- Opened: 3,193
- Closed: 1,977
Number of successful food establishments:
[tex]\[ \text{Food success} = \text{Opened} - \text{Closed} = 3,193 - 1,977 = 1,216 \][/tex]
#### Retail Establishments:
- Opened: 2,280
- Closed: 1,626
Number of successful retail establishments:
[tex]\[ \text{Retail success} = \text{Opened} - \text{Closed} = 2,280 - 1,626 = 654 \][/tex]
#### Service Establishments:
- Opened: 5,045
- Closed: 3,548
Number of successful service establishments:
[tex]\[ \text{Service success} = \text{Opened} - \text{Closed} = 5,045 - 3,548 = 1,497 \][/tex]
### Step 2: Calculate the Probabilities
Now, we need to calculate the probabilities for each situation.
#### Situation I: A food establishment succeeding and earning [tex]$50,000 or more For a food establishment to earn $[/tex]50,000 or more, the categories to consider are [tex]$50-75k$[/tex], [tex]$75-100k$[/tex], and over [tex]$100k$[/tex]. Summing up these categories:
[tex]\[ \text{Food} \geq \$50k = 601 + 258 + 114 = 973 \][/tex]
Probability:
[tex]\[ \text{Prob}_{\text{I}} = \frac{\text{Food} \geq \$50k}{\text{Food success}} \times 100 \][/tex]
[tex]\[ \text{Prob}_{\text{I}} = \frac{973}{1,216} \times 100 \approx 80.02\% \][/tex]
#### Situation II: A service establishment succeeding and earning between [tex]$25,000 and $[/tex]75,000
For a service establishment to earn between [tex]$25,000 and $[/tex]75,000, the categories to consider are [tex]$25-50k$[/tex] and [tex]$50-75k$[/tex]. Summing up these categories:
[tex]\[ \text{Service}_{\$25k-\$75k} = 739 + 432 = 1,171 \][/tex]
Probability:
[tex]\[ \text{Prob}_{\text{II}} = \frac{\text{Service}_{\$25k-\$75k}}{\text{Service success}} \times 100 \][/tex]
[tex]\[ \text{Prob}_{\text{II}} = \frac{1,171}{1,497} \times 100 \approx 78.22\% \][/tex]
#### Situation III: A retail establishment succeeding and earning no more than [tex]$50,000 For a retail establishment to earn no more than $[/tex]50,000, the categories to consider are up to [tex]$25k$[/tex] and [tex]$25-50k$[/tex]. Summing up these categories:
[tex]\[ \text{Retail} \leq \$50k = 813 + 548 = 1,361 \][/tex]
Probability:
[tex]\[ \text{Prob}_{\text{III}} = \frac{\text{Retail} \leq \$50k}{\text{Retail success}} \times 100 \][/tex]
[tex]\[ \text{Prob}_{\text{III}} = \frac{1,361}{654} \times 100 \approx 208.10\% \][/tex]
### Step 3: Determine Situations with at least 15% Probability
We need to determine which of the probabilities calculated are at least 15%:
1. [tex]\(\text{Prob}_{\text{I}} \approx 80.02\%\)[/tex] (greater than 15%)
2. [tex]\(\text{Prob}_{\text{II}} \approx 78.22\%\)[/tex] (greater than 15%)
3. [tex]\(\text{Prob}_{\text{III}} \approx 208.10\%\)[/tex] (greater than 15%)
Therefore, all three situations have a probability of at least 15%:
[tex]\[ \boxed{[1, 2, 3]} \][/tex]
Thus, based on the data, the correct answer is:
c. I, II, and III
