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Let's analyze the data and solve the problem.

### Given Table:
The table provides information about the order accuracy of four drive-thru restaurants (A, B, C, D):

| Drive-thru Restaurant | A    | B    | C    | D    |
|-----------------------|------|------|------|------|
| **Order Accurate**     | 340  | 269  | 232  | 143  |
| **Order Not Accurate** | 30   | 51   | 34   | 17   |

We are tasked with calculating the probability of getting two orders both from Restaurant D.

#### Total orders from Restaurant D:
- Accurate: 143
- Not Accurate: 17
- **Total from D** = 143 + 17 = **160**

### Part (a) - With Replacement:
We are asked to find the probability of selecting two orders from Restaurant D **with replacement**, and to determine if the events are independent.

#### Probability of choosing one order from Restaurant D:
\[ P(\text{D}) = \frac{\text{Total orders from D}}{\text{Total orders from all restaurants}} \]

The total number of orders across all restaurants is:
\[ 340 + 269 + 232 + 143 + 30 + 51 + 34 + 17 = 1116 \]

So,
\[ P(\text{D}) = \frac{160}{1116} \]

The probability of selecting two orders from Restaurant D **with replacement** is:
\[ P(\text{D and D}) = P(\text{D}) \times P(\text{D}) \]

Let me calculate this probability.The probability of selecting two orders from Restaurant D with replacement is approximately **0.0206**.

#### Are the events independent?
Yes, the events are independent because the probability of choosing the second order remains unaffected by the choice of the first order (since it's with replacement).

### Part (b) - Without Replacement:
Now, we need to calculate the probability of selecting two orders from Restaurant D **without replacement**, and determine if the events are independent.

#### Probability without replacement:
For the first selection, the probability is the same as above:
\[ P(\text{First D}) = \frac{160}{1116} \]

For the second selection, the total number of orders decreases by 1, and the number of orders from D also decreases by 1:
\[ P(\text{Second D after First D}) = \frac{159}{1115} \]

The total probability is:
\[ P(\text{D and D without replacement}) = P(\text{First D}) \times P(\text{Second D after First D}) \]

Let me calculate this as well.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can explain the process in detail if you want to calculate manually:

To find the probability without replacement:
1. Probability for the first order from Restaurant D is \( \frac{160}{1116} \).
2. For the second order, it becomes \( \frac{159}{1115} \), since both the total number of orders and orders from D decrease by 1.

The probability without replacement is:
\[ P(\text{D and D without replacement}) = \frac{160}{1116} \times \frac{159}{1115} \]

These events are **not independent**, because the outcome of the first selection affects the second.

If you have any questions or want further details, let me know!

### Five related questions:
1. How do we determine if events are independent?
2. How does replacement affect probability calculations?
3. Can you explain the difference between conditional and independent probability?
4. How would the probability change if there were more restaurants?
5. What is the importance of sample space in probability problems?

### Tip:
Always consider whether the problem asks for sampling with or without replacement—this is crucial in determining whether events are independent!

If two orders are selected, find the probability that they are both from Restaurant D. Assume that the selections are made with and without replacement. Are the events independent?

Learn how to calculate the probability of selecting two orders from Restaurant D, considering scenarios with and without replacement. This problem involves key probability concepts, including independent events and the effect of replacement on probability.

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