Say I am working with a set of the numbers $\{1,2,3,4,5\}$ a uniform sampling of the numbers could result in the sequence $1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,...$. Another uniform sampling could result in $1,3,5,2,4,1,3,5,2,4,1,3,5,2,4,...$. Neither of these are random because my process for choosing did not allow things like $2,1$ in either case. My process was the obvious one given the sequences.

Now, consider the following random sampling algorithm. Flip a coin. If it is heads, return $1$ and if it is tails, return $3$. A valid sequence of outputs might be $1,1,3,1,3,3,1,1,3,3,3,1,1,1,3,1,3,1,1,1,3,3,3$. Note that this is not uniform for the set $\{1,2,3,4,5\}$.

So this would suggest that there is no redundancy in saying "uniform and at random".

I would say that "random" refers to the process by which they are drawn, "uniform" refers to the property of the results of the process that each outcome is equally likely (distribution).

Say I am working with a set of the numbers $\{1,2,3,4,5\}$ a uniform sampling of the numbers could result in the sequence $1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,...$. Another uniform sampling could result in $1,3,5,2,4,1,3,5,2,4,1,3,5,2,4,...$. Neither of these are random because my process for choosing did not allow things like $2,1$ in either case. My process was the obvious one given the sequences.

Now, consider the following random sampling algorithm. Flip a coin. If it is heads, return $1$ and if it is tails, return $3$. A valid sequence of outputs might be $1,1,3,1,3,3,1,1,3,3,3,1,1,1,3,1,3,1,1,1,3,3,3$. For the set $\{1,2,3,4,5\}$ this is clearly not uniform.

So this would suggest that there is no redundancy in saying "uniform and at random".

I would say that "random" refers to the process by which they are drawn, "uniform" refers to the property of the results of the process that each outcome is equally likely (distribution).

Say I am working with a set of the numbers $\{1,2,3,4,5\}$ a uniform sampling of the numbers could result in the sequence $1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,...$. Another uniform sampling could result in $1,3,5,2,4,1,3,5,2,4,1,3,5,2,4,...$. Clearly, neither of these are random, however.

Now, consider the following random sampling algorithm. Flip a coin. If it is heads, return $1$ and if it is tails, return $3$. A valid sequence of outputs might be $1,1,3,1,3,3,1,1,3,3,3,1,1,1,3,1,3,1,1,1,3,3,3$. Note that this is not uniform.

So this would suggest that there is no redundancy in saying "uniform and at random".

I would say that "random" refers to the process by which they are drawn, "uniform" refers to the property of the results of the process that each outcome is equally likely.

Say I am working with a set of the numbers $\{1,2,3,4,5\}$ a uniform sampling of the numbers could result in the sequence $1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,...$. Another uniform sampling could result in $1,3,5,2,4,1,3,5,2,4,1,3,5,2,4,...$. Neither of these are random because my process for choosing did not allow things like $2,1$ in either case. My process was the obvious one given the sequences.

Now, consider the following random sampling algorithm. Flip a coin. If it is heads, return $1$ and if it is tails, return $3$. A valid sequence of outputs might be $1,1,3,1,3,3,1,1,3,3,3,1,1,1,3,1,3,1,1,1,3,3,3$. Note that this is not uniform for the set $\{1,2,3,4,5\}$.

So this would suggest that there is no redundancy in saying "uniform and at random".

I would say that "random" refers to the process by which they are drawn, "uniform" refers to the property of the results of the process that each outcome is equally likely (distribution).