To determine the probability that you will have to wait at least 10 minutes for the bus:
1. Understanding the scenario:
- A bus arrives at the bus stop every 15 minutes.
- You arrive at the bus stop at a random time.
2. Total time interval:
- The buses arrive every 15 minutes, so the total time interval is 15 minutes.
3. Time you have to wait:
- We are interested in the probability that you will have to wait at least 10 minutes for the bus.
- Therefore, the time you have to wait is 10 minutes within the 15-minute interval.
4. Calculate the probability:
- To find the probability that you will have to wait at least 10 minutes, think about the favorable condition with respect to the total time.
- If you arrive within the first 5 minutes of the 15-minute interval, you will wait at least 10 minutes.
- Therefore, the favorable time window is 5 minutes out of the total 15-minute window.
5. Probability calculation:
- The probability is calculated as the favorable time interval over the total time interval:
[tex]\[
\text{Probability} = \frac{\text{Favorable time interval}}{\text{Total time interval}} = \frac{5}{15}
\][/tex]
- Simplify the fraction:
[tex]\[
\frac{5}{15} = \frac{1}{3}
\][/tex]
Thus, the probability that you will have to wait at least 10 minutes for the bus is [tex]\(\frac{1}{3}\)[/tex].
Therefore, the correct answer is:
[tex]\(\boxed{\frac{1}{3}}\)[/tex]
