Part A: Calculate the empirical conditional probability of an employee who walks to work, given that they live under 2 miles away.
First, let’s extract the given data for employees who live under 2 miles away and determine their mode of transportation:
- Car: 15
- Public Transit: 20
- Bike: 30
- Walk: 25
We need to calculate the total number of employees who live under 2 miles away:
[tex]\[ \text{Total under 2 miles} = 15 (\text{Car}) + 20 (\text{Public Transit}) + 30 (\text{Bike}) + 25 (\text{Walk}) = 90 \][/tex]
Next, we focus on the number of employees who walk to work and live under 2 miles away. This value is:
[tex]\[ \text{Walk and under 2 miles} = 25 \][/tex]
The empirical conditional probability [tex]\(P(\text{Walk} | \text{Under 2 miles})\)[/tex] is the number of employees who walk and live under 2 miles away divided by the total number of employees living under 2 miles:
[tex]\[ P(\text{Walk} | \text{Under 2 miles}) = \frac{\text{Walk and under 2 miles}}{\text{Total under 2 miles}} = \frac{25}{90} \approx 0.2778 \][/tex]
So, the empirical conditional probability of an employee who walks to work, given that they live under 2 miles away, is approximately 0.2778.
Part B: Is an employee more or less likely to walk to work if they live under 2 miles away? Justify the answer mathematically.
To answer this, we need to compare the conditional probability of walking (given living under 2 miles away) to the overall probability of walking to work.
First, let’s calculate the total number of employees who walk to work:
[tex]\[ \text{Total walk} = 25 (\text{under 2 miles}) + 10 (\text{2-4 miles}) + 5 (\text{5-6 miles}) + 0 (\text{over 6 miles}) = 40 \][/tex]
Now, we calculate the overall probability [tex]\(P(\text{Walk})\)[/tex] of an employee walking to work:
[tex]\[ P(\text{Walk}) = \frac{\text{Total walk}}{\text{Total employees}} = \frac{40}{300} \approx 0.1333 \][/tex]
Now we compare the conditional probability [tex]\(P(\text{Walk} | \text{Under 2 miles})\)[/tex] with the overall probability [tex]\(P(\text{Walk})\)[/tex]:
[tex]\[ P(\text{Walk} | \text{Under 2 miles}) \approx 0.2778 \][/tex]
[tex]\[ P(\text{Walk}) \approx 0.1333 \][/tex]
Since [tex]\(P(\text{Walk} | \text{Under 2 miles})\)[/tex] (0.2778) is greater than [tex]\(P(\text{Walk})\)[/tex] (0.1333), an employee is more likely to walk to work if they live under 2 miles away.
