To determine the probabilities for the described scenarios, follow these steps:
### Step-by-Step Solution:
1. Summarize the Frequency Data:
- [tex]$\text{0-3 minutes: 9 customers}$[/tex]
- [tex]$\text{4-7 minutes: 10 customers}$[/tex]
- [tex]$\text{8-11 minutes: 12 customers}$[/tex]
- [tex]$\text{12-15 minutes: 4 customers}$[/tex]
- [tex]$\text{16-19 minutes: 4 customers}$[/tex]
- [tex]$\text{20-23 minutes: 2 customers}$[/tex]
- [tex]$\text{24-27 minutes: 2 customers}$[/tex]
2. Calculate the Total Number of Customers:
[tex]\[
\text{Total customers} = 9 + 10 + 12 + 4 + 4 + 2 + 2 = 43
\][/tex]
3. Calculate the Number of Customers Waiting at Least 12 Minutes:
- This includes customers in the intervals [tex]\(12-15\)[/tex], [tex]\(16-19\)[/tex], [tex]\(20-23\)[/tex], and [tex]\(24-27\)[/tex] minutes.
[tex]\[
\text{Customers waiting at least 12 minutes} = 4 + 4 + 2 + 2 = 12
\][/tex]
4. Calculate the Probability of Waiting at Least 12 Minutes:
[tex]\[
P(\text{waiting at least 12 minutes}) = \frac{\text{Customers waiting at least 12 minutes}}{\text{Total customers}} = \frac{12}{43} \approx 0.279
\][/tex]
5. Calculate the Number of Customers Waiting Between 8 and 15 Minutes:
- This includes customers in the intervals [tex]\(8-11\)[/tex] and [tex]\(12-15\)[/tex] minutes.
[tex]\[
\text{Customers waiting between 8 and 15 minutes} = 12 + 4 = 16
\][/tex]
6. Calculate the Probability of Waiting Between 8 and 15 Minutes:
[tex]\[
P(\text{waiting between 8 and 15 minutes}) = \frac{\text{Customers waiting between 8 and 15 minutes}}{\text{Total customers}} = \frac{16}{43} \approx 0.372
\][/tex]
7. Combine the Two Probabilities:
- These events are distinct (one is an interval within another), so probabilities can be added directly:
[tex]\[
P(\text{waiting at least 12 minutes or between 8 and 15 minutes}) = 0.279 + 0.372 \approx 0.558
\][/tex]
Therefore, the probability that a randomly selected customer waited at least 12 minutes or between 8 and 15 minutes is approximately [tex]\(0.558\)[/tex].
