Say you have a die with n number of sides. Assume the die is weighted properly and each side has an equal chance of coming up. How do I determine the minimum number of rolls needed so that results show an equal distribution, within an expected margin of error?

musicbachv7

musicbachv7

Answered question

2022-08-10

How do I determine sample size for a test?
Say you have a die with n number of sides. Assume the die is weighted properly and each side has an equal chance of coming up. How do I determine the minimum number of rolls needed so that results show an equal distribution, within an expected margin of error?
I assume there is a formula for this, but I am not a math person, so I don't know what to look for. I have been searching online, but haven't found the right thing.

Answer & Explanation

Ayla Coffey

Ayla Coffey

Beginner2022-08-11Added 8 answers

Step 1
Let there be N independent rolls. Let N i be the number of times outcome i has occured, and thus N 1 + + N n = N ..
The empirical distribution after N rolls is p i ^ = N i / N ,, for 1 i n , and let the actual distribution be denoted p i ( p i = 1 / n in your case). Let p = ( p 1 , , p n ) , p ^ = ( p 1 ^ , , p n ^ ) and define the total variation ( 1 ) distance between the two distributions as p p ^ 1 = i = 1 n p i p i ^ .
Step 2
One of the results in this domain which is easy to apply was proved by Devroye [1]:
Let ε 20 N / n then P [ p p ^ 1 > ε ] 3 exp ( N ε 2 / 25 ).
If you want to directly bound the maximum difference in probabilities, i.e., p p ^ = sup 1 i n p i p i ^ you can obtain P [ p p ^ > ε ] 4 exp ( N ε 2 / 2 ) , ε > 0..
Finally, note that if the second inequality is used with ε 0 , the corresponding ε in the first inequality can be as large as ε n ε 0 .

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