The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between 0 and 23 minutes. What is the probability that a person waits more than 8 minute 0.8261 0.1304
The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between 0 and 23 minutes. What is the probability that a person waits more than 8 minute 0.8261 0.1304
College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter9: Counting And Probability
Section: Chapter Questions
Problem 2P: Family Planning A intend to have two children. What is tie probability that they will have child of...
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Question
![## Problem Statement
The below problem examines uniform distribution and probability calculations for wait times. This example is highly relevant for understanding the concepts of probability in uniformly distributed scenarios.
**Problem:**
The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between 0 and 23 minutes. What is the probability that a person waits more than 8 minutes?
### Multiple Choice Answers:
- \( \mathbf{0.8261} \)
- \( \mathbf{0.1304} \)
- \( \mathbf{0.6522} \)
- \( \mathbf{0.3043} \)
- \( \mathbf{0.4783} \)
### Detailed Explanation:
Uniform distribution is used when each outcome within a given range is equally likely. In this example, the time a person waits for the bus ranges from 0 to 23 minutes, and each minute within this range is equally probable.
#### Steps to Solve:
1. **Calculate Total Range:**
- The total time range is from 0 to 23 minutes:
\[ \text{Total Range} = 23 - 0 = 23 \text{ minutes} \]
2. **Calculate the Desired Range:**
- We need to find the probability that a person will wait more than 8 minutes. Therefore, the desired range is from 8 to 23 minutes:
\[ \text{Desired Range} = 23 - 8 = 15 \text{ minutes} \]
3. **Calculate Probability:**
- The probability of waiting more than 8 minutes is the ratio of the desired range to the total range:
\[ P(\text{waiting more than 8 minutes}) = \frac{\text{Desired Range}}{\text{Total Range}} = \frac{15}{23} \approx 0.6522 \]
Hence, the probability that a person must wait more than 8 minutes for the bus is approximately \( 0.6522 \).
#### Answer:
\[ \boxed{0.6522} \]
Understanding uniform distributions and their applications in real-world scenarios helps solidify foundational probability concepts. This problem highlights how to transform probability theory into practical insights, particularly useful in operations research, transportation planning, and various service-oriented industries.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F49b6872f-8540-4d67-8f44-b13749eac041%2F7a6285ae-2ee1-480f-b577-72e90a238948%2Ft066qn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Problem Statement
The below problem examines uniform distribution and probability calculations for wait times. This example is highly relevant for understanding the concepts of probability in uniformly distributed scenarios.
**Problem:**
The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between 0 and 23 minutes. What is the probability that a person waits more than 8 minutes?
### Multiple Choice Answers:
- \( \mathbf{0.8261} \)
- \( \mathbf{0.1304} \)
- \( \mathbf{0.6522} \)
- \( \mathbf{0.3043} \)
- \( \mathbf{0.4783} \)
### Detailed Explanation:
Uniform distribution is used when each outcome within a given range is equally likely. In this example, the time a person waits for the bus ranges from 0 to 23 minutes, and each minute within this range is equally probable.
#### Steps to Solve:
1. **Calculate Total Range:**
- The total time range is from 0 to 23 minutes:
\[ \text{Total Range} = 23 - 0 = 23 \text{ minutes} \]
2. **Calculate the Desired Range:**
- We need to find the probability that a person will wait more than 8 minutes. Therefore, the desired range is from 8 to 23 minutes:
\[ \text{Desired Range} = 23 - 8 = 15 \text{ minutes} \]
3. **Calculate Probability:**
- The probability of waiting more than 8 minutes is the ratio of the desired range to the total range:
\[ P(\text{waiting more than 8 minutes}) = \frac{\text{Desired Range}}{\text{Total Range}} = \frac{15}{23} \approx 0.6522 \]
Hence, the probability that a person must wait more than 8 minutes for the bus is approximately \( 0.6522 \).
#### Answer:
\[ \boxed{0.6522} \]
Understanding uniform distributions and their applications in real-world scenarios helps solidify foundational probability concepts. This problem highlights how to transform probability theory into practical insights, particularly useful in operations research, transportation planning, and various service-oriented industries.
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