Math Problem Statement
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:
-
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). -
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. -
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. -
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: -
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:
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:
- What is the probability that a randomly selected person's wait time is less than 3 minutes?
- How would the probability change if the distribution was not uniform?
- Can you calculate the probability that the wait time is between 5 and 8 minutes?
- What if the wait times were distributed uniformly between 0 and 15 minutes—how would the probability change?
- 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
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