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Chapter 5: Problem 75

Four students must work together on a group project. They decide that each will take responsibility for a particular part of the project, as follows: Because of the way the tasks have been divided, one student must finish before the next student can begin work. To ensure that the project is completed on time, a time line is established, with a deadline for each team member. If any one of the team members is late, the timely completion of the project is jeopardized. Assume the following probabilities: 1\. The probability that Maria completes her part on time is 0.8 . 2\. If Maria completes her part on time, the probability that Alex completes on time is 0.9 , but if Maria is late, the probability that Alex completes on time is only \(0.6 .\) 3\. If Alex completes his part on time, the probability that Juan completes on time is 0.8 , but if Alex is late, the probability that Juan completes on time is only \(0.5 .\) 4\. If Juan completes his part on time, the probability that Jacob completes on time is \(0.9,\) but if Juan is late, the probability that Jacob completes on time is only 0.7 . Use simulation (with at least 20 trials) to estimate the probability that the project is completed on time. Think carefully about this one. For example, you might use a random digit to represent each part of the project (four in all). For the first digit (Maria's part), \(1-8\) could represent on time, and 9 and 0 could represent late. Depending on what happened with Maria (late or on time), you would then look at the digit representing Alex's part. If Maria was on time, \(1-9\) would represent on time for Alex, but if Maria was late, only \(1-6\) would represent on time. The parts for Juan and Jacob could be handled similarly.

Short Answer

Expert verified
After simulating the completion process 20 times with random digits representing each student's task completion status, we can estimate the probability of the project being completed on time. In this simulation, the estimated probability is approximately the number of successful trials divided by the total number of trials (20). This result may vary slightly with each simulation but should be close to the calculated probability of 0.5184 from step 1.

Step by step solution

01

Calculate the Probability if Everyone Completes On Time

Calculate the overall probability if everyone completes on time (Maria -> Alex -> Juan -> Jacob): \(P(\text{Project On Time}) = P(\text{Maria On Time}) P(\text{Alex On Time}\mid\text{Maria On Time}) P(\text{Juan On Time}\mid\text{Alex On Time}) P(\text{Jacob On Time}\mid\text{Juan On Time})\) Plug in the values from the problem: \(P(\text{Project On Time}) = (0.8)(0.9)(0.8)(0.9)\) \(P(\text{Project On Time}) = 0.5184\)
02

Simulate the Project Completion Process

For this step, we will simulate each student's completion 20 times and then estimate the probability. The process will involve using random digits to represent whether a student completes their tasks on time or late. 1. Maria - \(1 \thickspace to \thickspace 8\) - On Time, \(9 \thickspace and \thickspace 0 \) - Late 2. Alex (Maria On Time) - \(1 \thickspace to \thickspace 9\) - On Time, \(0 \) - Late 3. Alex (Maria Late) - \(1 \thickspace to \thickspace 6\) - On Time, \(7 \thickspace to \thickspace 0 \) - Late 4. Juan (Alex On Time) - \(1 \thickspace to \thickspace 8\) - On Time, \(9 \thickspace and \thickspace 0 \) - Late 5. Juan (Alex Late) - \(1 \thickspace to \thickspace 5\) - On Time, \(6 \thickspace to \thickspace 0 \) - Late 6. Jacob (Juan On Time) - \(1 \thickspace to \thickspace 9\) - On Time, \(0 \) - Late 7. Jacob (Juan Late) - \(1 \thickspace to \thickspace 7\) - On Time, \(8 \thickspace to \thickspace 0 \) - Late Example code for simulating the process: ``` import random n_trials = 20 successes = 0 for _ in range(n_trials): marias_digit = random.randint(0, 9) maria_on_time = marias_digit <= 8 if maria_on_time: alexs_digit = random.randint(0, 9) alex_on_time = alexs_digit <= 9 else: alexs_digit = random.randint(0, 9) alex_on_time = alexs_digit <= 6 if alex_on_time: juans_digit = random.randint(0, 9) juan_on_time = juans_digit <= 8 else: juans_digit = random.randint(0, 9) juan_on_time = juans_digit <= 5 if juan_on_time: jacobs_digit = random.randint(0, 9) jacob_on_time = jacobs_digit <= 9 else: jacobs_digit = random.randint(0, 9) jacob_on_time = jacobs_digit <= 7 if maria_on_time and alex_on_time and juan_on_time and jacob_on_time: successes += 1 estimated_probability = successes / n_trials ```
03

Estimate the Probability of On-Time Completion

After simulating 20 trials, divide the number of successful trials (all students on time) by the total number of trials (20) to estimate the probability of on-time completion: \(P(\text{Project On Time}) \approx \frac{\text{Number of Successful Trials}}{\text{Total Number of Trials}}\) We can use the example code from Step 2 to calculate the estimated probability: ``` estimated_probability = successes / n_trials ``` This will give us an estimated probability of on-time completion for the project. Note that the results may vary as you run the simulation multiple times, but they should be close to the calculated probability of 0.5184 from step 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability
Probability is the measure of the likelihood of an event occurring. In the context of a group project, it's important to understand the chances of each student completing their task on time. These probabilities help us anticipate potential delays and plan accordingly.

Each task completion has a probability assigned to it. For example, Maria has an 80% chance or a probability of 0.8 of finishing her part on time. This tells us that, out of 100 such projects, we expect Maria to be on time in 80 of them.
  • Probabilities range from 0 to 1.
  • A probability of 0 means the event never happens, while a probability of 1 means it always happens.
  • Probabilities can be used to make decisions and predictions in both everyday situations and complex projects.
These concepts form the backbone of planning in projects like these, helping teams adjust timelines and avoid bottlenecks.
Random Digit Method
The Random Digit Method is a technique often used in simulations to model random events and calibrate probabilities. In our group project example, it helps simulate the likelihood of students completing tasks on time.

By using random digits, we can create scenarios that reflect real-life possibilities. We allocate digit ranges to represent probability. For instance, digits 1 to 8 might signify Maria finishing on time (0.8 probability). Conversely, digits 9 and 0 might imply delay, matching her 0.2 probability of being late.
  • Random digits can range from 0 to 9, representing various outcomes.
  • This method allows repeated simulation trials, which improve accuracy of probability estimations.
  • Such simulations help visualize possible outcomes without needing real-world trials.
The Random Digit Method is valuable in teaching and forecasting, offering insight into the probabilistic realms of events and decision-making.
Conditional Probability
Conditional Probability is the likelihood of an event occurring given that another event has already occurred. It's crucial when one event's outcome affects another's probability, as seen in our project timeline scenario.

When Maria finishes on time, Alex has a 0.9 probability of being timely, but that drops to 0.6 if Maria is late. This dependent relationship shows how one task's outcome changes another's likelihood.
  • Conditional probability is represented mathematically as \( P(B|A) \), meaning the probability of B given A happened.
  • This notion is vital for events with interdependent stages, like supply chains or team tasks.
  • Understanding these relationships helps in crafting effective contingency plans.
In any sequence of tasks, recognizing and calculating conditional probabilities helps manage risks and prepare for alternate outcomes.
Group Projects
Group projects involve multiple individuals working collaboratively to achieve a common goal, with distinct components often attributing to unique team members. Such projects benefit significantly from the study of probability and simulation to foresee and mitigate potential delays.

In our example project, each student's performance impacts the overall outcome. Delays in one part may lead to setbacks in subsequent tasks, which is why setting timelines and understanding dependencies are crucial.
  • Effective communication is key to the success of group projects.
  • Assigning clear responsibilities can help each member understand their role.
  • Using probability and simulation allows teams to plan better and anticipate potential issues.
Group projects teach essential skills like time management, responsibility, and teamwork, all while fostering an environment where students can learn the math behind probability and its practical applications.

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Most popular questions from this chapter

According to The Chronicle for Higher Education Almanac (2016), there were 1,003,329 Associate degrees awarded by U.S. community colleges in the \(2013-2014\) academic year. A total of 613,034 of these degrees were awarded to women. a. If a person who received an Associate degree in \(2013-\) 2014 is selected at random, what is the probability that the selected person will be female? b. What is the probability that the selected person will be male?

A large cable company reports that \(42 \%\) of its customers subscribe to its Internet service, \(32 \%\) subscribe to its phone service, and \(51 \%\) subscribe to its Internet service or its phone service (or both). a. Use the given probability information to set up a hypothetical 1000 table. b. Use the table to find the following: i. the probability that a randomly selected customer subscribes to both the Internet service and the phone service. ii. the probability that a randomly selected customer subscribes to exactly one of the two services.

The article "A Crash Course in Probability" from The Economist included the following information: The chance of being involved in an airplane crash when flying on an Airbus 330 from London to New York City on Virgin Atlantic Airlines is 1 in \(5,371,369 .\) This was interpreted as meaning that you "would expect to go down if you took this flight every day for 14,716 years." The article also states that a person could "expect to fly on the route for 14,716 years before plummeting into the Atlantic." Comment on why these statements are misleading.

An online store offers two methods of shipping-regular ground service and an expedited 2 -day shipping. Customers may also choose whether or not to have the purchase gift wrapped. Suppose that the events \(E=\) event that the customer chooses expedited shipping \(G=\) event that the customer chooses gift wrap are independent with \(P(E)=0.26\) and \(P(G)=0.12\). a. Construct a hypothetical 1000 table with columns corresponding to whether or not expedited shipping was chosen and rows corresponding to whether or not gift wrap was selected. b. Use the information in the table to calculate \(P(E \cup G)\). Give a long- run relative frequency interpretation of this probability.

a. Suppose events \(E\) and \(F\) are mutually exclusive with \(P(E)=0.41\) and \(P(E)=0.23\). i. What is the value of \(P(E \cap F) ?\) ii. What is the value of \(P(E \cup F) ?\) b. Suppose that for events \(A\) and \(B, P(A)=0.26, P(B)=0.34\), and \(P(A \cup B)=0.47\). Are \(A\) and \(B\) mutually exclusive? How can you tell?
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