# Independence for chosen digits for binary numbers selection

Let us consider all binary $n$-digits numbers. Out of them, we choose at random $m$ distinct numbers. Denote by $Y_1,...,Y_n$ the number of the chosen numbers with the first digit 1, ..., the number of the chosen numbers with the $n$-th digit 1, respectively.

How can we prove that the random variables $Y_1,....,Y_n$ are independent?

I claim that $Y_1, \dots, Y_n$ are not independent. Let $n = 2$ so that we are sampling from the set $\{00, 01, 10, 11\}$. Let us sample $m = 2$ of these binary numbers without replacement. The set of all possible outcomes is thus \begin{align*} \Omega = \{\{00,01\}, \{00,10\}, &\{00,11\}, \\ \{01,10\}, &\{01,11\}, \\ &\{10, 11\}\} \end{align*}
Suppose we observe $Y_1 = 0$. Then we know immediately that our sample was $\{00, 01\}$, meaning that $Y_2$ must take on the value 1. So $$P(Y_2 = 1 | Y_1 = 0) = 1 \neq 2/3 = P(Y_2 = 1).$$ Since the conditinal and marginal probabilities are not equal, it follows that $Y_1$ and $Y_2$ are not independent.