# Random shuffle of numbers

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!

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