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**2. Electric trains on a certain line run every half hour between midnight and 5 in the morning. Identify the probability that a man entering the station at a random time during this period will have to wait at least 15 minutes.**

### Solution:

1. The man enters the station at a random time, and the trains run every **30 minutes**. Therefore, the maximum waiting time is 30 minutes.

2. To find the probability that the man will have to wait at least 15 minutes, we need to consider the length of the time interval he could arrive within a 30-minute window:
   - **Time interval** where the man waits at least 15 minutes: 15 minutes to 30 minutes.
   - **Total time window**: 30 minutes.

3. The time intervals for waiting at least 15 minutes is from **15 minutes to 30 minutes**, which is **15 minutes long**. The probability can be found as the ratio of the favorable time interval to the total time window:
   \[
   P(\text{wait at least 15 minutes}) = \frac{15}{30} = 0.5
   \]

Thus, the probability that the man will have to wait at least 15 minutes is **0.5** or **50%**.

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Would you like further clarification on this problem or any details? Here are a few related questions you could explore:

1. What is the probability that the man will wait less than 15 minutes?
2. How would the probability change if the trains ran every 20 minutes instead of 30?
3. What is the expected wait time for the man if he arrives at a random time?
4. If the waiting time follows a uniform distribution, what is the cumulative distribution function (CDF) for waiting times?
5. How would the problem change if the man arrives during a non-uniform train schedule?

### Tip:
For problems like this, always look at the total time period first, then identify the specific intervals of interest to calculate probabilities effectively.

Electric trains on a certain line run every half hour between midnight and 5 in the morning. Identify the probability that a man entering the station at a random time during this period will have to wait at least 15 minutes.

Learn how to calculate the probability of waiting at least 15 minutes for a train arriving every half hour. This problem covers probability and uniform distribution concepts, suitable for students in Grades 9-12.

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