Question: The two-way table below gives the thousands of commuters in Massachusetts in 2015 by transportation method and one-way length of commute.
Less than 15 minutes 15-29 minutes 30-44 minutes 45-59 minutes 60 or more minutes Total
Private vehicle 636 908 590 257 256 2647
Public Transportation 9 54 96 62 108 329
Other 115 70 23 7 7 222
Total 760 1032 709 326 371 3198
Given that the commuter used public transportation, find the probability that the commuter had a commute of 60 or more minutes.
P (60+ minutes | public transportation) =
1. **State the problem:** We want to find the conditional probability that a commuter had a commute of 60 or more minutes given that they used public transportation.
2. **Recall the formula for conditional probability:**
$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$
where $A$ is the event "commute of 60 or more minutes" and $B$ is the event "used public transportation".
3. **Identify values from the table:**
- Number of commuters who used public transportation and had a commute of 60 or more minutes: $108$ (thousands)
- Total number of commuters who used public transportation: $329$ (thousands)
4. **Calculate the conditional probability:**
$$P(60+ \text{ minutes} | \text{public transportation}) = \frac{108}{329}$$
5. **Simplify the fraction if possible:**
Since $108$ and $329$ have no common factors other than 1, the fraction is already in simplest form.
6. **Convert to decimal (optional):**
$$\frac{108}{329} \approx 0.3283$$
7. **Interpretation:**
Given that a commuter used public transportation, there is approximately a $32.83\%$ chance that their commute was 60 or more minutes.
**Final answer:**
$$P(60+ \text{ minutes} | \text{public transportation}) = \frac{108}{329} \approx 0.3283$$