Question

PROBLEM SOLVING You and 19 other students volunteer to present the "Best Teacher" award at a school banquet. One student volunteer will be chosen to present the award. Each student worked at least 1 hour in preparation for the banquet. You worked for 4 hours, and the group worked a combined total of 45 hours. For each situation, describe a process that gives you a "fair" chance to be chosen, and find the probability that you are chosen. a. "Fair" means equally likely. b. "Fair" means proportional to the number of hours each student worked in preparation.

Short answer

In Scenario A, where 'fair' means equally likely, your probability of being chosen is `1/20`. In Scenario B, where 'fair' means proportional to the number of hours each student worked in preparation, your probability of being chosen is `4/45`.

Step-by-step solution

Identifying participants

First, identify the total number of participants including you, which is 20.

Probability calculation for equally likely scenario

For situation 'a', a 'fair' chance implies an equally likely scenario, which means each student, regardless of how many hours they worked, has an equal opportunity to be selected. The probability is calculated by dividing 1 (your single chance) by the total number of opportunities, which is 20 (the total number of students). So, it is `1/20`.

Total Work calculation

For situation 'b', we need to first calculate the total amount of work done by you and your peers. According to the information provided, it is 45 hours.

Probability calculation for proportional scenario

Under this situation, a 'fair' chance implies the opportunity to be chosen is directly proportional to the amount of work done. This means the more hours a student worked, the higher probability they have to be selected. So, the probability for you to be selected is the amount of time you worked divided by the total hours worked by the group, i.e., `4/45`.

Key concepts

Equally Likely Outcomes

Understanding equally likely outcomes is crucial when it comes to calculating probabilities in situations where fairness is based on equal chances for all involved. By definition, outcomes are equally likely when each one has the same chance of occurring. In our textbook example, where students volunteer to present an award, granting each student an equally likely opportunity to be chosen irrespective of their contribution means that the individual efforts, in this case, the hours worked, do not factor into their probability of being chosen. Instead, each student has a 1 in 20 chance, which is represented mathematically by the fraction \( \frac{1}{20} \). This illustrates a fundamental principle in probability where the total number of equally possible outcomes determines an individual's chance at being selected.

When using the concept of equally likely outcomes to ensure fairness, it's like holding a lottery where each ticket has an identical shot at winning. It encapsulates the idea of egalitarian treatment, with a clear and straightforward method of selection—drawing a name from a hat with 20 names would be a perfect example. An equally likely approach doesn't account for merit or contribution, only the randomness of chance.

Proportional Chance

Moving on to the concept of a proportional chance, we look at a scenario where fairness is based on the proportional effort or contribution of the individuals involved. In layman's terms, think of a proportional chance as a 'the more you put in, the more you get out' scenario. In other words, it's a chance that's directly tied to how much work each student has put into the preparation for the banquet.

Under a proportional chance system, you calculate the probability of being chosen by dividing the amount of work an individual has done by the total amount of work done by everyone. Applying this to our example, where you've worked 4 hours out of a total of 45, we find your chance of being chosen is represented by \( \frac{4}{45} \), after summing all students' preparation time. This method ensures a form of equitable fairness, rewarding the individuals who have contributed more significantly. In such a system, the probability is not fixed but varies and is directly correlated to the individual input. It's akin to earning more raffle tickets for every hour worked and then being more likely to win a draw because you have more tickets than others.

Total Work Calculation

In the context of our textbook problem, total work calculation plays a significant role in determining a fair distribution of probability when using the proportional method. This requires summing up the entire contribution of all individuals involved. In the 'Best Teacher' award scenario, this means adding up all the hours that you and your fellow students have contributed towards the preparation of the event. The collective effort here is quantified as 45 hours of work.

To perform a total work calculation, one would typically list out each individual's hours of contribution and add them together, ensuring no work is left unaccounted. However, in this case, the total work is already given. To calculate an individual's proportional chance after this step, as stated before, you would then take your individual contribution (4 hours) and divide it by this total (45 hours). The accuracy of the total work calculation is imperative, as any discrepancy would directly affect the fairness and proportionality of the chance distribution. Remember, this method attributes a weight to each hour worked and thus recognizes the varied efforts of each participant.