Sure! Let's organize our solution step-by-step to get the requested probabilities and then order them from least to greatest.
### Step 1: Calculate the Probability of International Flight Given it's On Time
We need to find [tex]\( P(\text{International} \,|\, \text{On Time}) \)[/tex].
This is computed using the formula:
[tex]\[ P(\text{International} \,|\, \text{On Time}) = \frac{\text{Number of On Time International Flights}}{\text{Total Number of On Time Flights}} \][/tex]
Given data:
- On Time International Flights: 56
- Total On Time Flights: 198
Hence,
[tex]\[ P(\text{International} \,|\, \text{On Time}) = \frac{56}{198} \approx 0.2828 \][/tex]
### Step 2: Calculate the Probability of an On Time and Domestic Flight
We need to find [tex]\( P(\text{On Time and Domestic}) \)[/tex].
This is computed using the formula:
[tex]\[ P(\text{On Time and Domestic}) = \frac{\text{Number of On Time Domestic Flights}}{\text{Total Number of Flights}} \][/tex]
Given data:
- On Time Domestic Flights: 142
- Total Number of Flights: 300
Hence,
[tex]\[ P(\text{On Time and Domestic}) = \frac{142}{300} \approx 0.3238 \][/tex]
### Step 3: Calculate the Probability of a Delayed Domestic Flight
We need to find [tex]\( P(\text{Delayed} \,|\, \text{Domestic}) \)[/tex].
This is computed using the formula:
[tex]\[ P(\text{Delayed} \,|\, \text{Domestic}) = \frac{\text{Number of Delayed Domestic Flights}}{\text{Total Number of Domestic Flights}} \][/tex]
Given data:
- Delayed Domestic Flights: 68
- Total Number of Domestic Flights: 210
Hence,
[tex]\[ P(\text{Delayed} \,|\, \text{Domestic}) = \frac{68}{210} \approx 0.4733 \][/tex]
### Step 4: Order Probabilities From Least to Greatest
We have the following probabilities:
1. [tex]\( P(\text{International} \,|\, \text{On Time}) \approx 0.2828 \)[/tex]
2. [tex]\( P(\text{On Time and Domestic}) \approx 0.3238 \)[/tex]
3. [tex]\( P(\text{Delayed} \,|\, \text{Domestic}) \approx 0.4733 \)[/tex]
Now, let's order them from least to greatest:
[tex]\[ 0.2828, 0.3238, 0.4733 \][/tex]
### Final Answer:
[tex]\[ P(\text{International} \,|\, \text{On Time}) \][/tex]
[tex]\[ P(\text{On Time and Domestic}) \][/tex]
[tex]\[ P(\text{Delayed} \,|\, \text{Domestic}) \][/tex]
So, the correctly ordered probabilities from least to greatest are:
[tex]\[ 0.2828, 0.3238, 0.4733 \][/tex]
