Problem 213

[dnosrc]

Hi, are these expected values correct for a 3x3 matrix?:

Bell 9: 2.843364198
Bell 10: 3.704475309

EDIT:

no thats wrong

[elr]

is it possible to solve this problem by just using a simulation ?

does the expected number of empty squares for 4x4 grid and 50 rings is 5.68670000 (or something close to 5.6/7) ?

[TripleM]

First question: not telling

Second question: that's not correct, but is close.

[monkeybrain]

I may have found a neat way how to calculate the average number of empty squares, but I am getting a wrong answer. Could someone please verify the following results for grid size of 10x10:

Bell 1: 31.224826388889
Bell 2: 33.650095051246
Bell 3: 34.711160908625
Bell 4: 35.224091288067
Bell 5: 35.580623381056

Thank you! MB

[TripleM]

I'm afraid they're all too high.

[oleglyamin]

Could someone check my values, please?
Matrix 4x4.
Rounded to three decimal places.
Bell 1: 4.778
Bell 2: 2.504
Bell 3: 2.984
Bell 4: 2.912
Bell 5: 2.922

Thanks.

[thundre]

Could someone check my values, please?
Matrix 4x4.
Rounded to three decimal places.
Bell 1: 4.778
Bell 2: 2.504
Bell 3: 2.984
Bell 4: 2.912
Bell 5: 2.922

Your number for Bell 1 is right. After that, they should continue to increase toward a certain limit.

[pieisgood]

edit: NVM, solved it

[rcp2361]

For problem 213, for the 4x4 case, the expected number of zeros before the 1st bell is 4.777 when the initial state is all "1"s (not after the first bell). After the first bell; this number increases.

[lucchini]

Hi!
Could someone check my values for a 4x4 matrix, please?

Bell 1: 4.778
Bell 2: 5.369
Bell 3: 5.246
Bell 4: 5.359
Bell 5: 5.314

[mpiotte]

Hi!
Could someone check my values for a 4x4 matrix, please?

Bell 1: 4.778
Bell 2: 5.369
Bell 3: 5.246
Bell 4: 5.359
Bell 5: 5.314

Bell 1 is correct, the others are not.

[RolandMontpellier]

Hello, all of you
The latest post being more than 10 years old, I am not sure this one will be read, but let's try it anyway.
I don't succeed obtaining the right answer in this problem, and I would appreciate some tips. Here is what I did : I first tried a Monte Carlo method. It didn't work, but I could understand that the correct answer wasn't far from [removed by moderator] (I hope I don't spoil things too much writing this). I then wrote a first version of an algorithm, using float numbers, but it didn't work. Nevertheless, the answer once again was close to XXX. Therefore, I imagined that maybe I should use only integers. I wrote a third version. It was much slower than the previous one, and the difference between this answer and the previous one was so small that it affected only the last digit (the 12th one). And of course, the answer was still wrong.
Hoping again not to spoil things, I explain the way I have imagined my algorithm. In fact, I try to find, cell by cell, the probability that each flea can arrive at this cell after 50 bells. Thus, at the end, I can obtain the probability that this cell is occupied bye none of the beas. Adding all this probabilities, I should obtain the expected answer.
Why is this wrong ? Thanks in advance for your help.

[pjt33]

Roland, just a guess, but you may be treating random variables as independent when they aren't.

To debug, you could consider a smaller system (e.g. 3x3 or 4x4) which you can brute-force for comparison with the more sophisticated calculation.