91Ó°ÊÓ

Many cities regulate the number of taxi licenses, and there is a great deal of competition for both new and existing licenses. Suppose that a city has decided to sell 10 new licenses for \(\$ 25,000\) each. A lottery will be held to determine who gets the licenses, and no one may request more than three licenses. Twenty individuals and taxi companies have entered the lottery. Six of the 20 entries are requests for 3 licenses, 9 are requests for 2 licenses, and the rest are requests for a single license. The city will select requests at random, filling as many of the requests as possible. For example, the city might fill requests for \(2,3,1\), and 3 licenses and then select a request for \(3 .\) Because there is only one license left, the last request selected would receive a license, but only one. a. An individual has put in a request for a single license. Use simulation to approximate the probability that the request will be granted. Perform at least 20 simulated lotteries (more is better!). b. Do you think that this is a fair way of distributing licenses? Can you propose an alternative procedure for distribution?

Step by step solution

01

- Analyzing the request demands

First, you need to understand the demand for licenses. Among the 20 requests, 6 are for 3 licenses, 9 are for 2 licenses and the remaining (20-6-9=5) are for 1 license. This means there are a total of 6*3 + 9*2 + 5*1 = 35 requested licenses.

02

- Simulating the lottery

The city has only 10 licenses available. To simulate the lottery, you must randomly choose requests (without replacement) and give them licenses until all 10 are given. Note that a request can only receive the number of licenses they asked for or the remaining licenses, whichever is smaller.

03

- Counting successful requests

After each simulation, record if the individual with a request for a single license got his license or not. A 'success' indicates that the individual received a license.

04

- Estimating the probability

After completing at least 20 simulations, the approximate probability of the individual receiving a license can be estimated as the number of successful simulations divided by the total number of simulations.

05

- Discussing Fairness

The fairness of this method of distribution can be discussed by considering if this method is unbiased and if it equally considers the needs of all participants. It depends on personal understanding and interpretation.

06

- Suggesting an alternative

A possible alternative procedure for distribution can be proposed based on understanding of fairness and effectiveness of distribution. For instance, it might be fairer to distribute the licenses in such a way that it considers the number of licenses requested by each individual/company and the total number of requests.

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Key Concepts

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

Probability estimation

Probability estimation in statistics is used to approximate how likely an event is to happen. In the case of the taxi license lottery, we're interested in estimating how probable it is for an individual requesting a single license to be successful.
This is done by simulating the lottery process multiple times. Performing many (at least 20) simulated lotteries helps gather enough data to make an informed estimation.
In each simulation, you randomly choose from the pool of requests. After all licenses are distributed, you check if the individual requesting one license was successful. Counting how many times this happens gives you a measure of likelihood.
For probability estimation, the basic principle is:

• Conduct multiple trials, ensuring enough data for reliability.
• Count successful outcomes in these trials.
• Calculate probability as the ratio of successful outcomes to total trials.

This method provides an empirical probability based on generated data.

Lotteries

Lotteries are a common method for random selection, often used in situations where demand exceeds supply. In the taxi license scenario, the lottery determines who gets these limited licenses. The process is random, which means everyone has a fair shot, but outcomes depend on chance.
Simulating a lottery involves mimicking this random selection process. Each request is akin to a lottery ticket, and the selection is made without replacement. This ensures that once a request is fulfilled, it isn't considered again in that simulation.
Key points in conducting a lottery include:

• Randomly selecting entries until all resources (licenses) are allocated.
• Ensuring fairness by maintaining equal participation for all entries.
• Using simulations to understand potential outcomes and probabilities.

Lotteries aim to provide an unbiased means of distribution, though some argue whether it's the fairest method available.

Fair distribution

Fair distribution refers to allocating resources in a way that is considered just and unbiased. In our context, it means distributing licenses such that all participants feel the process is equitable.
The lottery method distributes randomly, which can sometimes seem unfair, especially if based solely on chance without considering the number of licenses requested versus available.
To ensure fair distribution, consider these strategies:

• Balance need and randomness: allocate initially by request proportions, then random for excess.
• Weighted lottery: increase chances based on fewer licenses being requested relative to need.
• Round-robin method: fulfill every request systematically, adjusting remaining allocations.

While there's no one-size-fits-all method, understanding the mix of fairness principles and practicality helps in creating a fairer distribution approach.