The waiting times between a subway departure schedule and the arrival of a passenger are uniformly distributed between 0 and 8 minutes. Find the probability that a randomly selected passenger has a waiting time greater than 2.25 minutes.

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Part 1

Find the probability that a randomly selected passenger has a waiting time greater than 2.25 minutes.
  
enter your response here ​(Simplify your answer. Round to three decimal places as​ needed.)

To solve this problem, let's use the concept of a uniform distribution.

The waiting time \( X \) for a passenger is uniformly distributed between 0 and 8 minutes, which we denote as:
\[
X \sim U(0, 8)
\]

In a uniform distribution, the probability density function \( f(x) \) is constant across the interval:
\[
f(x) = \frac{1}{b - a}
\]
where \( a = 0 \) and \( b = 8 \). Therefore:
\[
f(x) = \frac{1}{8 - 0} = \frac{1}{8}
\]

We need to find the probability that a randomly selected passenger has a waiting time greater than 2.25 minutes, which is \( P(X > 2.25) \).

1. First, find the length of the interval where \( X > 2.25 \), which is from 2.25 to 8.
2. Calculate this probability by dividing the length of this interval by the total interval length of the uniform distribution.

So,
\[
P(X > 2.25) = \frac{8 - 2.25}{8} = \frac{5.75}{8} 
\]

Now, simplifying:
\[
P(X > 2.25) = 0.71875
\]

Rounding to three decimal places, we get:
\[
P(X > 2.25) \approx 0.719
\]

### Answer
The probability that a randomly selected passenger has a waiting time greater than 2.25 minutes is approximately **0.719**.

Would you like a further breakdown of this solution or have any questions?

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1. What is the probability that a passenger waits between 3 and 7 minutes?
2. How would the probability change if the uniform distribution range was between 0 and 10 minutes?
3. What is the expected waiting time for a passenger under this distribution?
4. How would we calculate the probability of waiting exactly 5 minutes?
5. How would you represent the cumulative distribution function (CDF) for this scenario?

**Tip:** In a uniform distribution \( U(a, b) \), the probability of an event in a subinterval \([c, d]\) is simply the length of \([c, d]\) divided by the length of \([a, b]\).

The waiting times between a subway departure schedule and the arrival of a passenger are uniformly distributed between 0 and 8 minutes. Find the probability that a randomly selected passenger has a waiting time greater than 2.25 minutes.

Find the probability that a randomly selected passenger has a waiting time greater than 2.25 minutes given a uniform distribution between 0 and 8 minutes. This problem uses the concept of uniform distribution probability and involves calculating intervals. Suitable for students in Grades 10-12.

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