How to Calculate the Mean Fraction of Occupied Seats in a Row?

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Discussion Overview

The discussion revolves around calculating the mean fraction of occupied seats in a row when a specified number of kids (N=50) are seated with at least one empty seat between them. The participants explore the implications of different seating arrangements and the definitions of "filled" seats.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the clarity of the problem, asking whether the requirement is to leave one or more empty seats between kids and what "all seats are filled" means in this context.
  • Another participant suggests that if there are 1 or 2 seats empty between kids, the calculation of mean fraction of occupied seats would depend on the total number of seats available.
  • It is proposed that if there are unlimited seats, the calculation would require a probability distribution for the number of seats left between kids.
  • A participant presents calculations for mean fractions based on different scenarios of available seats, indicating that the mean fraction can vary depending on the number of empty seats and the arrangement of kids.
  • One participant asserts that if seating continues until all seats are filled, the mean fraction of empty seats would be zero, although this statement does not clarify the conditions under which this is true.

Areas of Agreement / Disagreement

Participants express differing interpretations of the problem, particularly regarding the definitions of empty seats and the conditions for calculating the mean fraction of occupied seats. No consensus is reached on the specifics of the seating arrangement or the implications for the mean calculation.

Contextual Notes

Participants highlight the need for clarity on the total number of seats, the rules governing empty seats, and the implications of these factors on the calculations. There are unresolved assumptions about the seating arrangement and the conditions for determining when seats are considered filled.

dumsek
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There are a set of kids (let's say N=50) asked to sit in a row of seats, leaving at least one empty seat between them until all seats are filled. At the end, how do I calculate mean of the fraction of occupied seats? What will be the input to calculate mean?
 
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I am not sure you have all the information needed.
Do they leave 1 seat empty, or perhaps more than 1?
What does it mean for all seats to be filled? Does that mean there is someone in the first seat and someone in either the last or first to last seat and lots of gaps in between?
Do you know either the total number of seats or the odds of skipping more than one seat?
The mean is similar to the expected value...think of the sum over all options: (# of occupied seats) x (probability of that number).
 
there must be at least one seat empty between any two kids. They can sit in anyway until all seat are taken.
 
So, there might be 1 or 2 seats empty, but if 3 empty seats, then not all seats are taken, since another kid could fit?
Or are there unlimited seats available, so between any two kids there might be an unlimited number of seats?
Ultimately, if you are asking to find a fraction of occupied seats, that will be # taken / # total.
If one seat available, mean fraction is 1.00.
If two seats available, mean fraction is .5
If 3 seats available, you have 2/3 full in 2 of 3 cases, and 1/3 in one of the three cases (first person sits in the 2nd seat), for a mean seat fraction of 5/9.
I am still not clear what your actual problem is, if it is one where the seats are the limiter, then it will end up being some function of seats available. If student numbers is the limiter, then it will be students/seats. If seats are unlimited, and seats between are unlimited, then you need to have some sort of probability distribution for how many seats are left between kids.
 
dumsek said:
There are a set of kids (let's say N=50) asked to sit in a row of seats, leaving at least one empty seat between them until all seats are filled. At the end, how do I calculate mean of the fraction of occupied seats? What will be the input to calculate mean?

If the seating continues "until all seats are filled" in the row, the mean fraction of empty seats will be zero.
 

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