Math Problem Statement

At a small regional airport, the wait times at airport security are evenly (uniformly) distributed between 0 and 10 minutes. Determine the probability that a randomly selected person's wait time is more than 7 minutes.

Solution

The problem describes a uniform distribution of airport security wait times between 0 and 10 minutes. The goal is to find the probability that a randomly selected person’s wait time is more than 7 minutes.

Step-by-step solution:

  1. Uniform Distribution:
    The wait times are uniformly distributed between 0 and 10 minutes. For a uniform distribution, the probability density function (PDF) is constant within the given range (0 to 10).

  2. Range of interest:
    The event we are interested in is the wait time being more than 7 minutes. So, we need to find the probability that the wait time falls between 7 and 10 minutes.

  3. Total range of possible times:
    The total range of wait times is 0 to 10 minutes, so the length of the interval is 10 minutes.

  4. Interval for the event (more than 7 minutes):
    The interval for wait times greater than 7 minutes is from 7 to 10 minutes. The length of this interval is: 107=3minutes10 - 7 = 3 \, \text{minutes}

  5. Probability calculation:
    For a uniform distribution, the probability of an event occurring in a specific interval is the length of that interval divided by the total length of the distribution's range. So, the probability that a person waits more than 7 minutes is: P(X>7)=310=0.30P(X > 7) = \frac{3}{10} = 0.30

Thus, the probability that a randomly selected person’s wait time is more than 7 minutes is 0.30.

Do you want more details or have any questions?

Here are 5 related questions to expand on this concept:

  1. What is the probability that a randomly selected person's wait time is less than 3 minutes?
  2. How would the probability change if the distribution was not uniform?
  3. Can you calculate the probability that the wait time is between 5 and 8 minutes?
  4. What if the wait times were distributed uniformly between 0 and 15 minutes—how would the probability change?
  5. How do uniform distributions differ from normal distributions in probability problems?

Tip:

For uniform distributions, probabilities are directly proportional to the length of the interval. This makes uniform distributions particularly straightforward to work with!

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Math Problem Analysis

Mathematical Concepts

Probability
Uniform Distribution

Formulas

P(X > a) = (b - a) / (b - c)

Theorems

Uniform Distribution Probability

Suitable Grade Level

Grades 9-12