Answer :
To solve this problem, we'll calculate the probabilities of specific events occurring based on Alan's data. The key events of interest are:
1. The bus being on time and the weather being rainy.
2. The bus being delayed and the weather being foggy.
3. The bus being delayed and the weather being sunny.
4. The bus being on time and the weather being snowy.
We'll also classify these events as dependent or independent. Let's go through each step in detail.
### 1. Probability that the bus is on time and it is rainy
From the table, we see that 20 times the bus was on time and the weather was rainy out of a total of 189 recorded events.
[tex]\[ P(\text{on time} \cap \text{rainy}) = \frac{\text{Number of on time and rainy events}}{\text{Total number of events}} \][/tex]
[tex]\[ P(\text{on time} \cap \text{rainy}) = \frac{20}{189} \approx 0.1058 \][/tex]
### 2. Probability that the bus is delayed and it is foggy
From the table, we see that 4 times the bus was delayed and the weather was foggy out of a total of 189 recorded events.
[tex]\[ P(\text{delayed} \cap \text{foggy}) = \frac{\text{Number of delayed and foggy events}}{\text{Total number of events}} \][/tex]
[tex]\[ P(\text{delayed} \cap \text{foggy}) = \frac{4}{189} \approx 0.0212 \][/tex]
### 3. Probability that the bus is delayed and it is sunny
From the table, we see that 15 times the bus was delayed and the weather was sunny out of a total of 189 recorded events.
[tex]\[ P(\text{delayed} \cap \text{sunny}) = \frac{\text{Number of delayed and sunny events}}{\text{Total number of events}} \][/tex]
[tex]\[ P(\text{delayed} \cap \text{sunny}) = \frac{15}{189} \approx 0.0794 \][/tex]
### 4. Probability that the bus is on time and it is snowy
From the table, we see that 5 times the bus was on time and the weather was snowy out of a total of 189 recorded events.
[tex]\[ P(\text{on time} \cap \text{snowy}) = \frac{\text{Number of on time and snowy events}}{\text{Total number of events}} \][/tex]
[tex]\[ P(\text{on time} \cap \text{snowy}) = \frac{5}{189} \approx 0.0265 \][/tex]
### Dependent and Independent Events
Events are considered dependent if the occurrence of one event affects the probability of the other event. In this scenario, the events involving the bus's arrival time and the weather conditions are likely dependent. This means that the probability of the bus being on time or delayed would be influenced by the weather conditions.
Thus, the dependent events are:
- On time and rainy
- Foggy and delayed
- Delayed and sunny
- Snowy and on time
There are no independent events listed in this particular problem, as the promptness of the bus likely relies on weather conditions.
### Conclusion
Thus, the probabilities and event classifications are:
1. [tex]\( P(\text{on time} \cap \text{rainy}) \approx 0.1058 \)[/tex]
2. [tex]\( P(\text{delayed} \cap \text{foggy}) \approx 0.0212 \)[/tex]
3. [tex]\( P(\text{delayed} \cap \text{sunny}) \approx 0.0794 \)[/tex]
4. [tex]\( P(\text{on time} \cap \text{snowy}) \approx 0.0265 \)[/tex]
#### Dependent Events
- On time and rainy
- Foggy and delayed
- Delayed and sunny
- Snowy and on time
#### Independent Events
- None in this context
1. The bus being on time and the weather being rainy.
2. The bus being delayed and the weather being foggy.
3. The bus being delayed and the weather being sunny.
4. The bus being on time and the weather being snowy.
We'll also classify these events as dependent or independent. Let's go through each step in detail.
### 1. Probability that the bus is on time and it is rainy
From the table, we see that 20 times the bus was on time and the weather was rainy out of a total of 189 recorded events.
[tex]\[ P(\text{on time} \cap \text{rainy}) = \frac{\text{Number of on time and rainy events}}{\text{Total number of events}} \][/tex]
[tex]\[ P(\text{on time} \cap \text{rainy}) = \frac{20}{189} \approx 0.1058 \][/tex]
### 2. Probability that the bus is delayed and it is foggy
From the table, we see that 4 times the bus was delayed and the weather was foggy out of a total of 189 recorded events.
[tex]\[ P(\text{delayed} \cap \text{foggy}) = \frac{\text{Number of delayed and foggy events}}{\text{Total number of events}} \][/tex]
[tex]\[ P(\text{delayed} \cap \text{foggy}) = \frac{4}{189} \approx 0.0212 \][/tex]
### 3. Probability that the bus is delayed and it is sunny
From the table, we see that 15 times the bus was delayed and the weather was sunny out of a total of 189 recorded events.
[tex]\[ P(\text{delayed} \cap \text{sunny}) = \frac{\text{Number of delayed and sunny events}}{\text{Total number of events}} \][/tex]
[tex]\[ P(\text{delayed} \cap \text{sunny}) = \frac{15}{189} \approx 0.0794 \][/tex]
### 4. Probability that the bus is on time and it is snowy
From the table, we see that 5 times the bus was on time and the weather was snowy out of a total of 189 recorded events.
[tex]\[ P(\text{on time} \cap \text{snowy}) = \frac{\text{Number of on time and snowy events}}{\text{Total number of events}} \][/tex]
[tex]\[ P(\text{on time} \cap \text{snowy}) = \frac{5}{189} \approx 0.0265 \][/tex]
### Dependent and Independent Events
Events are considered dependent if the occurrence of one event affects the probability of the other event. In this scenario, the events involving the bus's arrival time and the weather conditions are likely dependent. This means that the probability of the bus being on time or delayed would be influenced by the weather conditions.
Thus, the dependent events are:
- On time and rainy
- Foggy and delayed
- Delayed and sunny
- Snowy and on time
There are no independent events listed in this particular problem, as the promptness of the bus likely relies on weather conditions.
### Conclusion
Thus, the probabilities and event classifications are:
1. [tex]\( P(\text{on time} \cap \text{rainy}) \approx 0.1058 \)[/tex]
2. [tex]\( P(\text{delayed} \cap \text{foggy}) \approx 0.0212 \)[/tex]
3. [tex]\( P(\text{delayed} \cap \text{sunny}) \approx 0.0794 \)[/tex]
4. [tex]\( P(\text{on time} \cap \text{snowy}) \approx 0.0265 \)[/tex]
#### Dependent Events
- On time and rainy
- Foggy and delayed
- Delayed and sunny
- Snowy and on time
#### Independent Events
- None in this context
