The manager of a bank recorded the amount of time each customer spent waiting in line during peak business hours one Monday. The frequency table below summarizes the results.

| Waiting Time (minutes) | Number of Customers |
|------------------------|---------------------|
| 0-3 | 9 |
| 4-7 | 10 |
| 8-11 | 12 |
| 12-15 | 4 |
| 16-19 | 4 |
| 20-23 | 2 |
| 24-27 | 2 |

If we randomly select one of the customers represented in the table, what is the probability that the waiting time is at least 12 minutes or between 8 and 15 minutes?

A. 0.558
B. 0.093



Answer :

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].