Assume we have a deck of cards. We randomly pick 13 cards of this stack.

Am I correct in assuming the variance of this randomly chosen deck of card is 14?

Furthermore asume now we pick randomly 13 cards WITH replacement from a deck of 52 cards. Am I correct in assuming the variance of this can just be found by squaring each of the the possible sum (min 13, max 169) minus the EV squared?

I would recommend a bit more information be provided. Both in terms of the problem setup and in terms of what you have already tried.

It sounds like you are giving each card a numerical value (Ace=1, Deuce=2, ..., Ten=10, Jack=11, Queen=12, King=13). Correct?

It sounds like you are interested in the SUM of the 13 randomly chosen cards out of a standard 52-card deck under two different sampling regimes: with replacement and without replacement. Correct?

It sounds like you are interested in the VARIANCE of the sample mean of the sum described above in the two sampling regimes. Correct?

Are you supposed to derive the two variances in question from first principles or are you supposed to apply formulas that have been previously derived for the two cases?

What have you done so far?

Apologies for the lack of information.

The values: A = 1 ..... K = 13

Interested in the sum. With and without replacement.

Yes I am interested about the sample mean. But would that be any different, as on average each of the cards would be equally distributed/likely to be within that range?

Sorry I do not mean with what you mean by the fifth paragraph.

Sorry for the robotic answer so far haha.

To be honest I haven't done much yet. With replacement it seem easy to just add the potential squared sums up and then substract the square of the mean (7), assuming the 13 cards each have 1 value (so A up to K) in it. And then do the "usual" variance calculations. However when there is no replacement it just seems impossible without writing a computer program.

It is hard for me to provide any guidance without knowing more.

I presume you know about random variables, expected values, definition of variance, covariance, etc.

Can you show that the expected value of the sample sum is the same under sampling with replacement and without replacement?

Can you write down the formulas for the variances of the sample sum under sampling with replacement and without replacement?

Can you then solve the formula under sampling with replacement (this is straightforward)?

Can you then solve the formula under sampling without replacement (this is fairly difficult)?

Thanks a lot for your help.

This was for an interview, a question I might possible would have gotten. I just realized I am unable to go beyond the basic regarding statistics, and given the replacement ordeal with this thing it is gonna be quite complex. That being said, sorry to have wasted your time.