Answer :
To solve this problem, we need to calculate the probability of getting an order that is not accurate or an order from Restaurant C, and determine if these events are disjoint.
### Step-by-Step Solution
1. Sum of Total Orders:
- Restaurant A: 326 (Accurate) + 34 (Not Accurate) = 360 orders
- Restaurant B: 265 (Accurate) + 54 (Not Accurate) = 319 orders
- Restaurant C: 233 (Accurate) + 39 (Not Accurate) = 272 orders
- Restaurant D: 126 (Accurate) + 10 (Not Accurate) = 136 orders
- Total orders: 360 + 319 + 272 + 136 = 1087 orders
2. Total Number of Inaccurate Orders:
- Not Accurate from A: 34 orders
- Not Accurate from B: 54 orders
- Not Accurate from C: 39 orders
- Not Accurate from D: 10 orders
- Total not accurate: 34 + 54 + 39 + 10 = 137 orders
3. Total Orders from Restaurant C:
- Orders from C: 233 (Accurate) + 39 (Not Accurate) = 272 orders
4. Probability Calculations:
- Probability of an order being not accurate:
[tex]\[ \text{P(Not Accurate)} = \frac{\text{Total Not Accurate Orders}}{\text{Total Orders}} = \frac{137}{1087} \approx 0.126 \][/tex]
- Probability of an order being from Restaurant C:
[tex]\[ \text{P(From Restaurant C)} = \frac{\text{Total Orders from C}}{\text{Total Orders}} = \frac{272}{1087} \approx 0.250 \][/tex]
- Probability of an order being not accurate and from Restaurant C:
[tex]\[ \text{P(Not Accurate and From C)} = \frac{\text{Not Accurate Orders from C}}{\text{Total Orders}} = \frac{39}{1087} \approx 0.036 \][/tex]
5. Combined Probability (Not Accurate OR From Restaurant C):
Using the formula for the union of two events:
[tex]\[ \text{P(A or B)} = \text{P(A)} + \text{P(B)} - \text{P(A and B)} \][/tex]
Here, A is "Not Accurate" and B is "From Restaurant C".
[tex]\[ \text{P(Not Accurate or From C)} = \text{P(Not Accurate)} + \text{P(From Restaurant C)} - \text{P(Not Accurate and From C)} \][/tex]
Plugging the values:
[tex]\[ \text{P(Not Accurate or From C)} \approx 0.126 + 0.250 - 0.036 \approx 0.340 \][/tex]
The probability of getting an order from Restaurant C or an order that is not accurate is approximately 0.340 (to three decimal places).
### Determining if the Events are Disjoint
Two events are disjoint if they cannot happen simultaneously. In this context, disjoint events would mean that an order cannot be both from Restaurant C and not accurate at the same time.
Since it is possible for an order to be from Restaurant C and be not accurate (as we have 39 such cases), these events are not disjoint.
### Conclusion
- Probability Calculation:
[tex]\[ \text{The probability of getting an order that is not accurate or is from Restaurant C is } 0.340 \][/tex]
- Event Disjoint Status:
- The events are not disjoint because it is possible to get an order that is both from Restaurant C and not accurate.
### Step-by-Step Solution
1. Sum of Total Orders:
- Restaurant A: 326 (Accurate) + 34 (Not Accurate) = 360 orders
- Restaurant B: 265 (Accurate) + 54 (Not Accurate) = 319 orders
- Restaurant C: 233 (Accurate) + 39 (Not Accurate) = 272 orders
- Restaurant D: 126 (Accurate) + 10 (Not Accurate) = 136 orders
- Total orders: 360 + 319 + 272 + 136 = 1087 orders
2. Total Number of Inaccurate Orders:
- Not Accurate from A: 34 orders
- Not Accurate from B: 54 orders
- Not Accurate from C: 39 orders
- Not Accurate from D: 10 orders
- Total not accurate: 34 + 54 + 39 + 10 = 137 orders
3. Total Orders from Restaurant C:
- Orders from C: 233 (Accurate) + 39 (Not Accurate) = 272 orders
4. Probability Calculations:
- Probability of an order being not accurate:
[tex]\[ \text{P(Not Accurate)} = \frac{\text{Total Not Accurate Orders}}{\text{Total Orders}} = \frac{137}{1087} \approx 0.126 \][/tex]
- Probability of an order being from Restaurant C:
[tex]\[ \text{P(From Restaurant C)} = \frac{\text{Total Orders from C}}{\text{Total Orders}} = \frac{272}{1087} \approx 0.250 \][/tex]
- Probability of an order being not accurate and from Restaurant C:
[tex]\[ \text{P(Not Accurate and From C)} = \frac{\text{Not Accurate Orders from C}}{\text{Total Orders}} = \frac{39}{1087} \approx 0.036 \][/tex]
5. Combined Probability (Not Accurate OR From Restaurant C):
Using the formula for the union of two events:
[tex]\[ \text{P(A or B)} = \text{P(A)} + \text{P(B)} - \text{P(A and B)} \][/tex]
Here, A is "Not Accurate" and B is "From Restaurant C".
[tex]\[ \text{P(Not Accurate or From C)} = \text{P(Not Accurate)} + \text{P(From Restaurant C)} - \text{P(Not Accurate and From C)} \][/tex]
Plugging the values:
[tex]\[ \text{P(Not Accurate or From C)} \approx 0.126 + 0.250 - 0.036 \approx 0.340 \][/tex]
The probability of getting an order from Restaurant C or an order that is not accurate is approximately 0.340 (to three decimal places).
### Determining if the Events are Disjoint
Two events are disjoint if they cannot happen simultaneously. In this context, disjoint events would mean that an order cannot be both from Restaurant C and not accurate at the same time.
Since it is possible for an order to be from Restaurant C and be not accurate (as we have 39 such cases), these events are not disjoint.
### Conclusion
- Probability Calculation:
[tex]\[ \text{The probability of getting an order that is not accurate or is from Restaurant C is } 0.340 \][/tex]
- Event Disjoint Status:
- The events are not disjoint because it is possible to get an order that is both from Restaurant C and not accurate.
