Let's solve the problem using the provided information step by step.
Firstly, we need to understand the total number of orders for each restaurant and calculate the total number of orders overall.
Step 1: Calculate Total Orders for Each Restaurant
Restaurant A:
- Order Accurate: 325
- Order Not Accurate: 35
- Total Orders: 325 + 35 = 360
Restaurant B:
- Order Accurate: 266
- Order Not Accurate: 57
- Total Orders: 266 + 57 = 323
Restaurant C:
- Order Accurate: 239
- Order Not Accurate: 31
- Total Orders: 239 + 31 = 270
Restaurant D:
- Order Accurate: 135
- Order Not Accurate: 15
- Total Orders: 135 + 15 = 150
Step 2: Calculate the Total Number of Orders
Total orders (all restaurants combined):
- Total Orders = 360 (A) + 323 (B) + 270 (C) + 150 (D) = 1103
Step 3: Find the Probability of Selecting One Order from Restaurant D
- Probability of selecting one order from Restaurant D: [tex]\(P(D) = \frac{Total\ Orders\ D}{Total\ Orders} = \frac{150}{1103} \approx 0.1359 \)[/tex]
Step 4: Calculate the Probability of Both Orders Being from Restaurant D
### a. Selections Are Made With Replacement
With replacement, the selections are independent. The probability of selecting a second order from Restaurant D is unaffected by the first selection.
- Probability (Both orders from D with replacement):
[tex]\[ P(D, D) = P(D) \times P(D) = 0.1359 \times 0.1359 \approx 0.0185 \][/tex]
The probability of getting two orders from Restaurant D with replacement is approximately 0.0185. The events are independent because choosing the first order does not affect the probability of the choice of the second order.
### b. Selections Are Made Without Replacement
Without replacement, the selections are dependent. The probability of selecting a second order from Restaurant D is affected by the first selection.
- Probability (First order from D): [tex]\( P(D_1) = \frac{150}{1103} \approx 0.1359 \)[/tex]
After selecting one order from D, there are 149 orders left from D and 1102 orders in total.
- Probability (Second order from D): [tex]\( P(D_2) = \frac{149}{1102} \approx 0.1353 \)[/tex]
- Probability (Both orders from D without replacement):
[tex]\[ P(D, D) = P(D_1) \times P(D_2) = 0.1359 \times 0.1353 \approx 0.0184 \][/tex]
The probability of getting two orders from Restaurant D without replacement is approximately 0.0184. The events are not independent because choosing the first order affects the probability of the choice of the second order.
### Summary:
- Probability with replacement: 0.0185
- Events with replacement are independent.
- Probability without replacement: 0.0184
- Events without replacement are not independent.
If two orders are selected, the probability that they are both from Restaurant [tex]\(D\)[/tex] is (with replacement) 0.0185. The events are independent because choosing the first order does not affect the probability of the choice of the second order.
