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There is a group of N unique things. A simple random sample sub group x can be taken without replacement where the number of samples is x, x < N and x = 25% of N. The sample sub group x is replaced after recording. How many simple random samples of size x need to be performed to see 90% of the unique items in group N?

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This sounds like a homework question, so I'm a bit weary of providing the answer, but maybe I can help you think about it.

I'm going to assume you're looking for the expected number of samples to exceed 90% of observation.

After sample 1 you will have observed 25% of the things. In sample 2, the probability of observing any individual thing is 0.25, so the expected number of new observations you will make is the proportion of items you haven't yet observed times the probability that each is observed. Using this you should be able to set up a recursive function to calculate the expected proportion observed after each sample.

Also, I suggest changing how you tagged this question, because this is not statistics, it's probability.

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  • $\begingroup$ Thank you for the response Alton. Not a homework question. Just trying to remember stuff from 10+ years ago. Your response was great! I can't remember the correct way to represent the recursive equation here but I am multiplying the number of items yet to be selected by .25 to find how many new items get selected adding this to the previous selected total and finally calculating my percentage. Much appreciated! $\endgroup$ – Bryce Aug 16 '18 at 21:13
  • $\begingroup$ Sounds like you've got it figured out! To write it properly, one way is to define $Y_i$ as the proportion of the things you've observed up until and including observation $i$. Your recursive function for $E[Y_i]$, the expected number of things you've observed up to observation $i$, would then be $E[Y_i] = E[Y_{i-1}] + 0.25(1- E[Y_{i-1}])$, with the initial condition that $E[Y_0]=0$. $\endgroup$ – Alton Aug 16 '18 at 22:40

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