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Let's solve this problem step-by-step.

### Given Information
The elevator arrival time is uniformly distributed between 10 and 30 seconds, so the waiting time \( X \) is modeled by a **Uniform distribution** \( U(10, 30) \). 

For a uniform distribution:
- The probability density function (PDF) is given by \( f(x) = \frac{1}{b - a} \) for \( a \leq x \leq b \).
- Here, \( a = 10 \) and \( b = 30 \), so \( f(x) = \frac{1}{30 - 10} = \frac{1}{20} = 0.05 \).

### Problem (a)
**Find the probability of waiting at most 25 seconds.**

This translates to finding \( P(X \leq 25) \).

1. Calculate the range of waiting time up to 25 seconds: \( 25 - 10 = 15 \) seconds.
2. The probability \( P(X \leq 25) \) is the area under the PDF from 10 to 25:
   \[
   P(X \leq 25) = f(x) \times \text{range} = 0.05 \times 15 = 0.75
   \]

So, for part (a), the probability is:
\[
p = 0.75
\]

### Problem (b)
**Find the probability of waiting between 15 and 25 seconds.**

This translates to finding \( P(15 \leq X \leq 25) \).

1. Calculate the range of waiting time between 15 and 25 seconds: \( 25 - 15 = 10 \) seconds.
2. The probability \( P(15 \leq X \leq 25) \) is the area under the PDF from 15 to 25:
   \[
   P(15 \leq X \leq 25) = f(x) \times \text{range} = 0.05 \times 10 = 0.5
   \]

So, for part (b), the probability is:
\[
p = 0.5
\]

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If you would like further clarification or details, let me know! 

### Additional Questions
1. What is the probability of waiting exactly 20 seconds?
2. How would the probability change if the elevator arrived uniformly between 5 and 30 seconds?
3. Can we calculate the mean and variance for this uniform distribution?
4. How would the probabilities change if the distribution was not uniform?
5. What is the probability of waiting more than 28 seconds?

### Tip:
In a uniform distribution, probabilities for ranges can be calculated as the length of the interval times the probability density, which is the inverse of the interval's total range.

In a certain hotel, the elevator takes between 10 and 30 seconds to arrive at a guest's floor once they've pushed the button. This means that the elevator arrival is uniformly distributed between 10 and 30 seconds once you've hit the button. (a) Find the probability of waiting at most 25 seconds. (b) Find the probability of waiting between 15 and 25 seconds.

Explore how to calculate the probability of waiting for an elevator arrival time modeled by a uniform distribution. This problem includes step-by-step solutions for determining the probability of waiting at most 25 seconds and between 15 and 25 seconds, utilizing concepts of uniform distribution and probability density.

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