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 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 \(\mathrm{Alex}\) 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

The short answer can't be defined in this case as each simulation would output a different probability. However, by following the proposed steps, the probability can be easily estimated.

Step-by-step solution

Understand the problem

In this problem, the probability of each student completing their task on time depends on the performance of the previous student. Maria's probability is independent while others depend on the person just before them. We need to simulate this scenario with at least 20 trials to estimate the probability that the project is completed on time.

Initialize the variable for successful trials

First, let's start by initializing a variable successful_trials to 0. This variable will keep track of the number of trials where all students complete the project on time.

Conduct the simulations

Run a loop for 20 trials. Inside each trial, generate a random number for each student's part. If Maria's number falls within 1-8, mark her as on time. Otherwise, she is late. This determines the chances for Alex. If Maria is on time and if Alex's number falls within 1-9, mark him as on time. If Maria is late, Alex's chances drop to 1-6. Repeat the process for Juan and Jacob. If all four students are on time, increment successful_trials by 1.

Calculate the estimated probability

After all trials have been conducted, divide the successful_trials by the total number of trials (20 in this case). This fraction will represent the estimated probability of the project being completed on time.

Key concepts

Conditional Probability

Conditional probability is a fundamental concept in probability theory that describes the likelihood of an event occurring given that another event has already happened. In educational settings, it's a critical concept for students to understand, as it applies to a multitude of real-world situations. Take, for instance, our exercise, which involves a sequence of tasks that need to be completed on time by different students for a group project. The completion of each task depends on the completion of the prior one, making this a classic example of conditional probability.

When we say that Alex has a 0.9 probability of finishing on time if Maria does, but only a 0.6 if she's late, we're talking about conditional probability. The probability of Alex finishing on time changes based on the outcome of Maria's task. To express this in mathematical terms, we define the conditional probability of event B given event A as P(B|A) = P(A and B) / P(A), provided that P(A) is not zero. In the context of our exercise, P(A) could be the probability of Maria finishing on time, while P(B|A) would be the probability of Alex finishing on time given that Maria has finished on time.

Probability Estimation

Probability estimation involves determining the likelihood of an event through experiments or simulations, rather than theoretical calculations. This is particularly useful when dealing with complex scenarios where multiple factors are at play, making analytical solutions difficult. In our textbook problem, we use probability estimation via simulation to predict the chance the entire project is completed on time.

To estimate this probability, we conduct a series of trials—in our case, at least 20. During each trial, we use a random digit to simulate the completion of each student’s part of the project. By tallying the trials where all students finish on time and dividing by the total number of trials, we get an empirical estimate of the probability. This hands-on approach not only makes the learning experience more interactive but also ingrains the principles of probability into students' understanding through repetitive practice.

Probability Dependence

The idea of probability dependence is crucial when events are not isolated but rather influenced by the outcomes of previous events. In the provided exercise, the probability of each student completing their part on time depends on whether the preceding student did so. This interconnection is what we refer to as probability dependence.

In simple terms, if the probability of an event changes when another event occurs, the two events are dependent. For example, Juan's probability of being on time is 0.8 if Alex is on time and 0.5 if Alex is late. Therefore, Juan's timely completion is dependent on Alex's performance. This concept is central to understanding many processes in nature, technology, and even social dynamics where outcomes are interconnected. Our educational exercises aim to convey how to handle dependent probabilities in a straightforward and intuitive manner, often using simulations to illustrate these relationships with tangible examples.