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The problem states that the waiting times between subway departures are uniformly distributed between 0 and 7 minutes, and you are asked to find the probability that a randomly selected passenger has a waiting time greater than 2.25 minutes.

### Step-by-Step Solution:
1. **Define the problem**:
   - The waiting time is uniformly distributed between 0 and 7 minutes. This is a uniform distribution problem, where the probability density function (PDF) is constant.
   - The total range is from 0 to 7 minutes.
   - You want the probability that the waiting time is greater than 2.25 minutes.

2. **The formula for uniform distribution**:
   The probability that \( X \) (waiting time) is greater than a certain value \( x \) is:
   \[
   P(X > x) = \frac{\text{Length of interval where } X > x}{\text{Total length of interval}}
   \]
   In this case:
   \[
   P(X > 2.25) = \frac{7 - 2.25}{7 - 0} = \frac{4.75}{7}
   \]

3. **Calculate the probability**:
   \[
   P(X > 2.25) = \frac{4.75}{7} \approx 0.679
   \]

### Final Answer:
The probability that a randomly selected passenger has a waiting time greater than 2.25 minutes is approximately **0.679** (rounded to three decimal places).

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Do you need more details or have any questions?

Here are 5 related questions to explore further:
1. What is the probability that a passenger has a waiting time less than 2.25 minutes?
2. How do we compute the mean and variance for a uniform distribution?
3. What is the probability that the waiting time is exactly 2.25 minutes?
4. If the waiting time range changed to 0 to 10 minutes, how would the result differ?
5. Can we calculate the median waiting time for this uniform distribution?

**Tip**: For uniform distributions, probabilities for exact values (like \( P(X = 2.25) \)) are always zero, since the probability is spread evenly across an interval!

The waiting times between a subway departure schedule and the arrival of a passenger are uniformly distributed between 0 and 7 minutes. Find the probability that a randomly selected passenger has a waiting time greater than 2.25 minutes.

Learn how to calculate the probability that a randomly selected passenger has a waiting time greater than 2.25 minutes. This problem uses the concept of uniform distribution, which is evenly spread over a range of values. Suitable for high school students studying probability.

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