What is the Expected Wheel Tax Revenue for the City in November?

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Discussion Overview

The discussion revolves around calculating the expected wheel tax revenue for a city based on the number of registered automobiles and the distribution of birthdays among the population. The focus is on the application of probability and expected value in this context, particularly for the month of November.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant presents the initial problem and proposes a formula for expected value, suggesting that the expected revenue can be calculated as E(X) = (74,506*50)*1/12, assuming uniform distribution of birthdays.
  • Another participant agrees with the need for a uniform distribution assumption and questions the expected revenue for February, indicating a potential need for clarification on the calculation method.
  • A participant responds that the expected revenue for February would be the same as for November using the same approach.
  • Another participant suggests that each day of the month should be considered equally likely, implying a more granular approach to the calculation.
  • One participant recalculates the expected revenue for November using a daily distribution, arriving at E(X) = (74,506*50)*30/365, and provides a similar calculation for February, assuming a non-leap year.
  • A later reply confirms the calculation for a non-leap year and prompts further exploration of the expected revenue for an average year.

Areas of Agreement / Disagreement

Participants generally agree on the need to assume a uniform distribution of birthdays for the calculations. However, there are differing views on the method of calculating expected revenue, particularly regarding the treatment of days in a month and the implications for different months.

Contextual Notes

The discussion does not resolve the assumptions regarding the distribution of birthdays or the implications of leap years on the calculations, leaving these aspects open for further exploration.

Who May Find This Useful

Individuals interested in probability, statistics, or municipal finance may find this discussion relevant, particularly those looking to understand the application of expected value in real-world scenarios.

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1. A city has 74,806 registered automobiles. Each is required to display a bumper decal showing that the owner paid an annual wheel tax of $50. By law, new decals needed to be purchased during the month of the owner’s birthday. How much wheel tax revenue can the city expect to receive in November?



2. expected value = E(X) = summation(x*f(x))
Leibnitz's rule



3. I thought there was not enough information in the problem. Can we assume that people are equally likely to be born in the different months of the year?
Anyway, all I have is E(X) = (74,506*50)*1/12
where the part in parenthesis is x and 1/12 is the probability function
(not sure).
 
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I think you are right that you need to assume a uniform distribution of birthdays, and I would write that assumption out explicitly.

Just one question about your answer - what is the expected revenue in (say) February?
 
Going by the same approach, it would be the same answer. Did you have something else in mind?
 
I think each day is equally likely...
 
Ah, I see. So for November, E(X) = (74,506*50)*30/365
and for February, assuming it's not a leap year, it would be E(X) = (74,506*50)*28/365

Thanks for the help.
 
That is the right answer for a non-leap year - can you figure it out for an average year?
 

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