Question

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 schedule 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 \(.8\). 2\. If Maria completes her part on time, the probability that Alex completes on time is \(.9\), but if Maria is late, the probability that Alex completes on time is only . 6 . 3\. If Alex completes his part on time, the probability that Juan completes on time is \(.8\), but if Alex is late, the probability that Juan completes on time is only .5. 4\. If Juan completes his part on time, the probability that Jacob completes on time is \(.9\), but if Juan is late, the probability that Jacob completes on time is only \(.7 .\) 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

The probability of timely completion of the project can be estimated from the simulation run. The exact probability would depend on the output of the simulation and can be calculated as explained in the last step.

Step-by-step solution

Define the Simulation

Set the simulation up: Assign a list of 20 random numbers ranging from 1 to 10 to each student. A digit in the range of 1-8 represents an on time completion for Maria, and the rest signifies a late completion. Subsequent digit ranges depends on her state (on time or late), which applies to Alex, then Juan, and finally Jacob.

Run the Simulation for Maria's Task

Analyze Maria's list of random numbers. Assign another list to denote 'on time' if the digit is between 1 and 8, and 'late' for 9 and 0.

Run the Simulation for Alex's Task

Depending on whether Maria is late or on time, assign on time for Alex if the digit on Alex's list is between 1 and 9 or 1 and 6 respectively. Otherwise, assign late.

Run the Simulation for Juan's Task

Depending on whether Alex is late or on time, assign on time for Juan if the digit on Juan's list is between 1 and 8 or 1 and 5 respectively. Otherwise, assign late.

Run the Simulation for Jacob's Task

Depending on whether Juan is late or on time, assign on time for Jacob if the digit on Jacob's list is between 1 and 9 or 1 and 7 respectively. Otherwise, assign late.

Determine Probability

Calculate the probability that the project is completed on time. This is the number of trials in which all tasks were completed on time divided by the total number of trials (20)

Key concepts

Conditional Probability

Conditional probability is a critical concept in understanding complex probabilistic scenarios, such as the teamwork project described in our exercise. It refers to the probability of an event occurring given that another event has already taken place.

For example, in the context of the exercise, the probability that Alex completes his part on time depends on whether Maria has completed her part on time. This conditional relationship between their completion times drastically affects the overall project's success. The simulation steps indicate that if Maria is on time, Alex has a 0.9 chance of being on time. However, if Maria is late, Alex's chance decreases to 0.6.

Understanding and accurately calculating conditional probabilities is essential, as it allows us to make better predictions and decisions in uncertain environments, just like in the project deadlines exercise. When dealing with conditional probabilities, it's important to consider the dependency between events, as failing to do so could lead to incorrect assessments of risk and likelihood.

Probabilistic Models

Probabilistic models are frameworks used to represent complex systems and processes in terms of probability. They provide a structured way to analyze random events and predict outcomes. The exercise provided showcases a probabilistic model in action through the simulation of the group project completion.

In this model, each student's ability to complete their part on time is expressed as a probability, with the outcome influenced by preceding events. This reflects real-life scenarios where outcomes are not purely deterministic but have elements of uncertainty.

Setting up a proper simulation as described in the solution steps allows us to estimate the likelihood of the project being completed on time. Probabilistic models like this are not just academic exercises; they're used in industries ranging from finance to engineering for risk assessment, decision making, and planning under uncertainty.

Statistical Analysis

Statistical analysis involves collecting and scrutinizing every data component to draw conclusions. In the project completion scenario, simulation with at least 20 trials is suggested to approximate the probability of timely project completion.

This recommendation is based on the law of large numbers, which stipulates that as more trials are conducted, the simulated results will converge to the expected value. During the statistical analysis phase of the simulation, the results from each of the 20 trials are aggregated to calculate the overall probability of the project being completed on time.

This analysis helps in understanding the behavior of the system and, ultimately, in making informed decisions based on the likelihood of different outcomes. It is a vital tool in many fields that deal with uncertainties and probabilities.