Answer :
To solve this problem, we need to calculate a couple of probabilities using the given table data. Let's break down the solution step by step:
1. List the given values:
- Orders from Restaurant A: Accurate = 331, Not Accurate = 31
- Orders from Restaurant B: Accurate = 271, Not Accurate = 51
- Orders from Restaurant C: Accurate = 246, Not Accurate = 33
- Orders from Restaurant D: Accurate = 128, Not Accurate = 13
2. Calculate total orders for each restaurant:
- Total orders from Restaurant A: [tex]\(331 + 31 = 362\)[/tex]
- Total orders from Restaurant B: [tex]\(271 + 51 = 322\)[/tex]
- Total orders from Restaurant C: [tex]\(246 + 33 = 279\)[/tex]
- Total orders from Restaurant D: [tex]\(128 + 13 = 141\)[/tex]
3. Calculate total accurate and not accurate orders:
- Total accurate orders: [tex]\(331 + 271 + 246 + 128 = 976\)[/tex]
- Total not accurate orders: [tex]\(31 + 51 + 33 + 13 = 128\)[/tex]
4. Calculate overall total orders:
- Overall total orders: [tex]\(362 + 322 + 279 + 141 = 1104\)[/tex]
5. Find the probability of getting an order that is not accurate:
[tex]\[ \text{Probability of not accurate order} = \frac{\text{Total not accurate orders}}{\text{Overall total orders}} = \frac{128}{1104} \approx 0.116 \][/tex]
6. Find the probability of getting an order from Restaurant C:
[tex]\[ \text{Probability of order from Restaurant C} = \frac{\text{Total orders from Restaurant C}}{\text{Overall total orders}} = \frac{279}{1104} \approx 0.253 \][/tex]
7. Find the number of not accurate orders from Restaurant C:
- Not accurate orders from Restaurant C: [tex]\(33\)[/tex]
8. Find the probability of getting an order that is not accurate and from Restaurant C:
[tex]\[ \text{Probability of not accurate order from Restaurant C} = \frac{\text{Not accurate orders from Restaurant C}}{\text{Overall total orders}} = \frac{33}{1104} \approx 0.030 \][/tex]
9. Calculate the probability of getting an order that is not accurate or is from Restaurant C (using the inclusion-exclusion principle):
[tex]\[ \text{Probability of not accurate or Restaurant C} = \text{Probability of not accurate} + \text{Probability of Restaurant C} - \text{Probability of not accurate and Restaurant C} \][/tex]
[tex]\[ = 0.116 + 0.253 - 0.030 = 0.339 \][/tex]
10. Determine if the events are disjoint:
- Events are disjoint if they cannot happen at the same time. Since the probability of getting an order that is not accurate and from Restaurant C is not zero (0.030), these events are not disjoint.
So, the probability of getting an order that is not accurate or is from Restaurant C is [tex]\(0.339\)[/tex].
The events of selecting an order that is not accurate and selecting an order from Restaurant C are not disjoint.
[tex]\(\boxed{0.339}\)[/tex]
1. List the given values:
- Orders from Restaurant A: Accurate = 331, Not Accurate = 31
- Orders from Restaurant B: Accurate = 271, Not Accurate = 51
- Orders from Restaurant C: Accurate = 246, Not Accurate = 33
- Orders from Restaurant D: Accurate = 128, Not Accurate = 13
2. Calculate total orders for each restaurant:
- Total orders from Restaurant A: [tex]\(331 + 31 = 362\)[/tex]
- Total orders from Restaurant B: [tex]\(271 + 51 = 322\)[/tex]
- Total orders from Restaurant C: [tex]\(246 + 33 = 279\)[/tex]
- Total orders from Restaurant D: [tex]\(128 + 13 = 141\)[/tex]
3. Calculate total accurate and not accurate orders:
- Total accurate orders: [tex]\(331 + 271 + 246 + 128 = 976\)[/tex]
- Total not accurate orders: [tex]\(31 + 51 + 33 + 13 = 128\)[/tex]
4. Calculate overall total orders:
- Overall total orders: [tex]\(362 + 322 + 279 + 141 = 1104\)[/tex]
5. Find the probability of getting an order that is not accurate:
[tex]\[ \text{Probability of not accurate order} = \frac{\text{Total not accurate orders}}{\text{Overall total orders}} = \frac{128}{1104} \approx 0.116 \][/tex]
6. Find the probability of getting an order from Restaurant C:
[tex]\[ \text{Probability of order from Restaurant C} = \frac{\text{Total orders from Restaurant C}}{\text{Overall total orders}} = \frac{279}{1104} \approx 0.253 \][/tex]
7. Find the number of not accurate orders from Restaurant C:
- Not accurate orders from Restaurant C: [tex]\(33\)[/tex]
8. Find the probability of getting an order that is not accurate and from Restaurant C:
[tex]\[ \text{Probability of not accurate order from Restaurant C} = \frac{\text{Not accurate orders from Restaurant C}}{\text{Overall total orders}} = \frac{33}{1104} \approx 0.030 \][/tex]
9. Calculate the probability of getting an order that is not accurate or is from Restaurant C (using the inclusion-exclusion principle):
[tex]\[ \text{Probability of not accurate or Restaurant C} = \text{Probability of not accurate} + \text{Probability of Restaurant C} - \text{Probability of not accurate and Restaurant C} \][/tex]
[tex]\[ = 0.116 + 0.253 - 0.030 = 0.339 \][/tex]
10. Determine if the events are disjoint:
- Events are disjoint if they cannot happen at the same time. Since the probability of getting an order that is not accurate and from Restaurant C is not zero (0.030), these events are not disjoint.
So, the probability of getting an order that is not accurate or is from Restaurant C is [tex]\(0.339\)[/tex].
The events of selecting an order that is not accurate and selecting an order from Restaurant C are not disjoint.
[tex]\(\boxed{0.339}\)[/tex]
