- We have arbitrarily many lottery scratch-n-win tickets
- Each ticket has a $p=0.1\%$ chance to win 1 prize (a success event)
Scenario A, Scratch n tickets and record the number of prizes won $X$: From the binomial distribution, $X$ will have a mean of 1 if $n=1000$. But, I'm more interested in the other scenario,
Scenario B, Scratch until we win ONE prize, then stop and record the nummber of tickets used as parameter $n$: I understand that the mean value of $n$ is going to be 1000 based on the fact that the expected value, 1, is equal to $np$.
But what distribution will n be described by if we make repetitions of scenario B? What I ultimately want is to be able to produce $n \pm Z$ with $95\%$ confidence where $n$ indicates the number of tickets used whereby 1 and only 1 prize is won. With which model can I get $n$ and $Z$ accurately?
So far the Normal approximation doesn't seem to be working well and I suspect it is due to $p$ being too close to 0.
