A manager of an apartment store reports that the time of a customer on the second floor must wait for the elevator has a uniform distribution ranging from 0  to 6 minutes.

If it takes the elevator 15 seconds to go from floor to floor, find the probability that a hurried customer can reach the first floor in less than 0.25 minutes after pushing the elevator button on the second floor.

Let's break down the problem:

1. **Uniform Distribution**: The waiting time for the elevator on the second floor is uniformly distributed between 0 and 6 minutes. This means each waiting time between 0 and 6 minutes is equally likely.

2. **Elevator Travel Time**: It takes the elevator 15 seconds to go from one floor to another. Since 15 seconds is equivalent to 0.25 minutes, the elevator will take 0.25 minutes to go from the second floor to the first floor.

3. **Total Time Constraint**: The hurried customer wants to reach the first floor in less than 0.25 minutes after pressing the button. This total time (waiting time + travel time) needs to be less than 0.25 minutes.

### Step-by-Step Solution

1. Let \( W \) represent the waiting time for the elevator on the second floor. Since the distribution of \( W \) is uniform from 0 to 6, the probability density function (PDF) is:
   \[
   f(w) = \frac{1}{6} \quad \text{for } 0 \leq w \leq 6
   \]

2. For the customer to reach the first floor in less than 0.25 minutes, the waiting time \( W \) must satisfy:
   \[
   W + 0.25 < 0.25
   \]
   Simplifying this, we get:
   \[
   W < 0
   \]

3. Since \( W \) cannot be negative (it's defined between 0 and 6), it's impossible for the waiting time to satisfy this condition. Therefore, there is **no chance** that the customer can reach the first floor in less than 0.25 minutes after pushing the button, given the elevator's travel time and the uniform distribution of the waiting time.

### Conclusion

The probability that a hurried customer can reach the first floor in less than 0.25 minutes after pushing the button is:
\[
\boxed{0}
\]

Would you like further clarification on any part of this solution? 

Here are some related questions to consider:

1. What is the probability that the customer will reach the first floor within 1 minute?
2. How does the probability change if the uniform distribution changes to a different range?
3. If the travel time between floors is reduced, how would that impact the probability?
4. How would the solution change if the waiting time follows a different distribution, like exponential?
5. What would be the expected waiting time for the elevator on the second floor?

**Tip:** Uniform distribution problems often require examining the range and calculating probability over a fixed interval.

A manager of an apartment store reports that the time a customer on the second floor must wait for the elevator has a uniform distribution ranging from 0 to 6 minutes. If it takes the elevator 15 seconds to go from floor to floor, find the probability that a hurried customer can reach the first floor in less than 0.25 minutes after pushing the elevator button on the second floor.

This solution explores the probability that a customer, waiting for an elevator with a uniform distribution of 0 to 6 minutes, can reach the first floor within 0.25 minutes after pressing the button. Learn how to apply uniform distribution properties and calculate probabilities with real-life scenarios. Suitable for high school students in Grades 11-12.

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