How is this not a counter-example to the law of large numbers?

Question

Let $\Omega = \{0,1\}$ and $X: \Omega \rightarrow \{0,1\}$ be a random variable s.t. $X = id$ with $E[X] = 0.5$ (i.e., $P(0) = 0.5 = P(1)$).

Let $X_1$, $X_2$, $\ldots$ be a sequence of i.i.d. random variables with $E[X_1] = E[X_2] = \ldots = 0.5 = E[X]$. In particular, let each of $X_i = id = X$.

Assertion: It seems to me necessarily the case that

$$ \overline{X}_n = {1 \over n} (X_1 + \ldots + X_n) = X $$

This is since

$$ {1 \over n} (X_1(0) + \ldots + X_n(0)) = {1 \over n}0 = 0 = X(0) $$

and

$$ {1 \over n} (X_1(1) + \ldots + X_n(1)) = {1 \over n}n = 1 = X(1) $$

But this means that as $n \rightarrow \infty$, it is not the case that $\overline{X}_n \to E[X] = 0.5$.

Question: How is this not a counter-example to the law of large numbers?

Answer 1

The law of large numbers applies to independent identically distributed random variables, whereas you're trying to apply it to identical random variables, which are not independent.