When can we reasonably assume a sequence of r.v.'s is IID in real life scenarios?
My question is based off the following example from Wasserman's All of Statistics:
Suppose we test a prediction method, a neural net for example, on a set of $n$ new cases. Let $X_i=1$ if the predictor is wrong and $X_i=0$ if the predictor is right. Then $\bar{X}_n = n^{-1}\sum_{i=1}^n X_i$ is the observed error rate. Each $X_i$ may be regarded as a Bernoulli with unknown mean $p$. Intuitively, we expect that $\bar{X}_n$ should be close to $p$. How likely is $\bar{X}_n$ to be within $\epsilon$ of $p$? We have that $V(\bar{X}_n)=V(X_1)/n=p(1-p)/n$ and
$$P(|\bar{X}_n-p|>\epsilon \leq V(\bar{X}_n/\epsilon^2) = p(1-p)/n\epsilon^2 \leq 1/4n\epsilon^2$$
where the last equation follows from Chebyshev's inequality.
In the above example, the variance of the sum of Bernoulli variables is calculated by assuming they are IID. Why is it reasonable to make this assumption?
