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, nine are requests for 2 licenses, and the rest are requests for a single license. The city will select requests at random, filling as much of the request 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 who wishes to be an independent driver 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

Identify the Objective

The objective is to use simulation to approximate the probability that the independent driver's request for a single license will be granted and also to provide a thoughtful opinion about the fairness of license distribution and propose an alternative.

02

Initial Setup

Firstly, set up initial condition. Array of size \(20\) will be created containing the various requests in accordance with the given conditions: six requests for 3 licenses, nine requests for 2 licenses and five requests for a single license. The independent driver's request will be uniquely identified, like 'ID' for example.

03

Start Simulation

Perform at least 20 simulation lotteries. For each simulation, shuffle the array of requests, then grant licenses in the order of shuffled requests, decrementing the total number of licenses by the current request amount each time, until the licenses run out. If the 'ID' request is granted a license, record this trial as a successful one.

04

Calculate Probability

After all simulations are complete, the approximate probability that the request will be granted can be found by dividing the number of successful trials by the total number of simulations.

05

Discuss Fairness and Propose an Alternative

Discuss whether the current method of distributing licenses is fair or not. An alternative method can be proposed, based on understanding of fairness and considering practical constraints.

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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 that a particular event will occur. In the context of this exercise, probability helps us understand the chances of an independent driver obtaining a single taxi license through a random lottery.

The probability \(P\) of an event is calculated as the number of successful outcomes divided by the total number of possible outcomes. To estimate the probability of the independent driver getting a license, we perform simulations.

By conducting 20 or more lottery simulations, where results favoring the driver's request are counted, we approximate the probability. The more simulations we run, the more accurate our approximation tends to become.

The formula: \[ P = \frac{\text{Number of Successful Trials}}{\text{Total Number of Simulations}} \]

Using this method offers a practical way to assess likelihood in complex real-world scenarios where analytical probability may be hard to calculate.

Independent Events

Independent events are those whose outcomes do not affect each other.

In the given exercise, each selection in the lottery is an independent event. Whether one person is awarded a license does not influence another person's chance of getting one.

This independence implies that each set of lottery results is as likely as any other, given the same conditions.

It is essential to understand this concept when setting up a simulation. Each shuffle and selection is independent, maintaining fairness and randomness in the distribution process.

Understanding independent events assures us that repeated simulations will yield unbiased insights into the probability of the independent driver's request being successful.

Random Selection

Random selection is the process of choosing participants in a way that each has an equal chance of being picked.

In our exercise, random selection is critical to ensure fairness in the license distribution process.

The simulation involves shuffling an array of requests randomly, representing the randomness of drawing names from a lottery pool. By doing so, each participant's request, including the independent driver, stands a chance of being selected without bias.

Random selection is valuable because it mimics real-life scenarios where choices or outcomes are not influenced by prior events or knowledge. This approach is crucial for obtaining valid probability estimates and ensuring the fairness of the allocation process in practical applications.

Fair Distribution Methods

Fair distribution methods aim to allocate resources or opportunities in an equitable manner. The lottery system used in this exercise attempts to distribute licenses fairly among taxi drivers and companies.

However, the fairness of such methods can be questioned, especially if all parties do not have equal needs or if the system inadvertently favors frequent occurrences of large requests.

One alternative to the current distribution could be a weighted lottery, where requests for fewer licenses might be given a higher probability to ensure smaller operators have fair access to licenses.

Considerations for alternative methods include:

• Weighted probabilities based on the size or need of the request.
• Capping the number of licenses that large requests can receive first.
• Implementing multiple rounds of distribution to prioritize unfulfilled requests.

By examining these alternatives, we can identify more refined methods to improve the fairness of the allocation process, helping everyone have a reasonable chance for success.