How to calculate confidence interval for dice hits? There is a dice with e edges [1;e]. The dice

vacinammo288

vacinammo288

Answered question

2022-04-23

How to calculate confidence interval for dice hits?
There is a dice with e edges [1;e]. The dice rolls in a one experiment r times. Thus the edge numbered 1 can be generated in this experiment from 0 to r times. I need to calculate theoretical 95% confidence interval for this count.

Answer & Explanation

Elsaidrge

Elsaidrge

Beginner2022-04-24Added 11 answers

It appears that you want what is called a prediction interval (PI, not "confidence interval") for Nr, where the random variable Nr is the total number of rolls that produce a 1 (out of r rolls). Now
Nr=X1++Xr, where
Xi={1, if the ith roll is a 10, otherwise
and we have the normal approximation when r is sufficiently large:
Nr=Binomial(r,p)Normal(μ,σ2)
where 

μ=E[Nr]=rp

σ2=V[Nr]=rp(1p)

p=P[Xi=1]=1e
(e being the number of sides on the die, which is assumed to be "fair").
NB:
E[Xi]=P[Xi=1] 1+P[Xi=0] 0 =p 1+(1p) 0 =p  V[Xi]=P[Xi=1](1E[Xi])2+P[Xi=0](0E[Xi])2 =p(1p)2+(1p)(0p)2 =p(1p)  E[Nr]=E[X1+...+Xr] =r E[X1] =r p  V[Nr]=V[X1+...+Xr] =V[X1]+...+V[Xr] because these are independent =r V[X1] =r p(1p)
An approximate 95% PI for Nr is then
μ±1.96σ
because, P[μ1.96σ<Nr<μ+1.96σ]0.95.
For your example, e=3, r=1000:  μ ± 1.96σ=333.3 ± 29.2[304,363]

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