Answer :
Let's address the problem step-by-step, incorporating the provided data and final answer.
### Step 1: Understanding the Data
The table gives us the number of accurate and non-accurate orders for four different restaurants:
[tex]\[ \begin{tabular}{|l|c|c|c|c|} \hline & A & B & C & D \\ \hline Order Accurate & 326 & 272 & 231 & 135 \\ \hline Order Not Accurate & 32 & 53 & 35 & 12 \\ \hline \end{tabular} \][/tex]
### Step 2: Total Orders Calculation
To find the total number of orders for each restaurant, we add the number of accurate and not accurate orders:
- For Restaurant A: [tex]\( 326 + 32 = 358 \)[/tex]
- For Restaurant B: [tex]\( 272 + 53 = 325 \)[/tex]
- For Restaurant C: [tex]\( 231 + 35 = 266 \)[/tex]
- For Restaurant D: [tex]\( 135 + 12 = 147 \)[/tex]
So, the total orders from each restaurant are:
[tex]\[ \{ 'A': 358, 'B': 325, 'C': 266, 'D': 147 \} \][/tex]
### Step 3: Total Orders from All Restaurants
Next, we sum these totals to get the total number of orders from all restaurants:
[tex]\[ 358 + 325 + 266 + 147 = 1096 \][/tex]
### Step 4: Probability of Selecting an Order from Restaurant D
The probability of selecting a single order from Restaurant D is:
[tex]\[ \frac{147}{1096} \approx 0.1341 \][/tex]
### Step 5: Probability of Two Orders from Restaurant D with Replacement
When selections are made with replacement, the events are independent. The probability of selecting two orders from Restaurant D with replacement is:
[tex]\[ P(\text{D and D with replacement}) = 0.1341 \times 0.1341 \approx 0.0180 \][/tex]
### Step 6: Probability of Two Orders from Restaurant D without Replacement
When selections are made without replacement, the events are dependent. The probability of selecting two orders from Restaurant D without replacement is:
[tex]\[ P(\text{D and D without replacement}) = \frac{147}{1096} \times \frac{146}{1095} \approx 0.0179 \][/tex]
### Summarizing the Results:
a. The probability of getting two orders from Restaurant D with replacement is [tex]\( 0.017989270872182852 \)[/tex], rounded to four decimal places, [tex]\( 0.0180 \)[/tex]. The events are independent because choosing the first order does not affect the probability of choosing the second order.
b. The probability of getting two orders from Restaurant D without replacement is [tex]\( 0.017883211678832115 \)[/tex], rounded to four decimal places, [tex]\( 0.0179 \)[/tex]. The events are not independent because choosing the first order affects the probability of choosing the second order.
### Filling in the Blanks:
The probability of getting two orders from Restaurant [tex]\( D \)[/tex] is [tex]\( 0.0180 \)[/tex] with replacement, and the events _are_ independent because choosing the first order _does not affect_ the probability of the choice of the second order.
The probability of getting two orders from Restaurant [tex]\( D \)[/tex] is [tex]\( 0.0179 \)[/tex] without replacement, and the events _are not_ independent because choosing the first order _affects_ the probability of the choice of the second order.
### Step 1: Understanding the Data
The table gives us the number of accurate and non-accurate orders for four different restaurants:
[tex]\[ \begin{tabular}{|l|c|c|c|c|} \hline & A & B & C & D \\ \hline Order Accurate & 326 & 272 & 231 & 135 \\ \hline Order Not Accurate & 32 & 53 & 35 & 12 \\ \hline \end{tabular} \][/tex]
### Step 2: Total Orders Calculation
To find the total number of orders for each restaurant, we add the number of accurate and not accurate orders:
- For Restaurant A: [tex]\( 326 + 32 = 358 \)[/tex]
- For Restaurant B: [tex]\( 272 + 53 = 325 \)[/tex]
- For Restaurant C: [tex]\( 231 + 35 = 266 \)[/tex]
- For Restaurant D: [tex]\( 135 + 12 = 147 \)[/tex]
So, the total orders from each restaurant are:
[tex]\[ \{ 'A': 358, 'B': 325, 'C': 266, 'D': 147 \} \][/tex]
### Step 3: Total Orders from All Restaurants
Next, we sum these totals to get the total number of orders from all restaurants:
[tex]\[ 358 + 325 + 266 + 147 = 1096 \][/tex]
### Step 4: Probability of Selecting an Order from Restaurant D
The probability of selecting a single order from Restaurant D is:
[tex]\[ \frac{147}{1096} \approx 0.1341 \][/tex]
### Step 5: Probability of Two Orders from Restaurant D with Replacement
When selections are made with replacement, the events are independent. The probability of selecting two orders from Restaurant D with replacement is:
[tex]\[ P(\text{D and D with replacement}) = 0.1341 \times 0.1341 \approx 0.0180 \][/tex]
### Step 6: Probability of Two Orders from Restaurant D without Replacement
When selections are made without replacement, the events are dependent. The probability of selecting two orders from Restaurant D without replacement is:
[tex]\[ P(\text{D and D without replacement}) = \frac{147}{1096} \times \frac{146}{1095} \approx 0.0179 \][/tex]
### Summarizing the Results:
a. The probability of getting two orders from Restaurant D with replacement is [tex]\( 0.017989270872182852 \)[/tex], rounded to four decimal places, [tex]\( 0.0180 \)[/tex]. The events are independent because choosing the first order does not affect the probability of choosing the second order.
b. The probability of getting two orders from Restaurant D without replacement is [tex]\( 0.017883211678832115 \)[/tex], rounded to four decimal places, [tex]\( 0.0179 \)[/tex]. The events are not independent because choosing the first order affects the probability of choosing the second order.
### Filling in the Blanks:
The probability of getting two orders from Restaurant [tex]\( D \)[/tex] is [tex]\( 0.0180 \)[/tex] with replacement, and the events _are_ independent because choosing the first order _does not affect_ the probability of the choice of the second order.
The probability of getting two orders from Restaurant [tex]\( D \)[/tex] is [tex]\( 0.0179 \)[/tex] without replacement, and the events _are not_ independent because choosing the first order _affects_ the probability of the choice of the second order.
