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Suppose I have $N$ distinct numbers labeled $n_1, \cdots,n_N$. I want to obtain a random shuffle of the $N$ numbers. Are the following statements equivalent?

  1. $P$(card i at position j) = $1/N$
  2. $P$(any permutations of the N cards) = $1/N!$

I can show 2 implies 1, but does 1 imply 2? When we say random shuffle, do we mean 2? Thanks!

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On its own, statement (1) does not imply statement (2). It describes only the marginal distribution of the card at each position $j$, not the joint distribution. In particular, there is nothing in statement (1) that prevents repitition of an individual card in multiple positions. To see that this probability statement does not imply (2), let $\mathbf{x} = (x_1,...,x_N)$ denote the vector of values, and consider the following joint distribution, which is consistent with (1) but not with (2):

$$\mathbb{P}(x_1 = x_2 = \cdots = x_N = n_i) = \frac{1}{N} \quad \quad \quad \text{for all } i=1,...,N.$$

This joint distribution says that you draw one card at random and it occupies every position. It leads to the marginal distribution in (1), but it is obviously inconsistent with the joint distribution specified in (2). Unlike (2), it is not a proper specification of "simple random sampling without replacement".

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