A binomial distribution is the distribution of the number of successes of n independent, identical Bernoulli trials. What happens when the trials are dependent and the Bernoulli trials are not identical by which I mean that the probability of success from trial to trial varies? How identical and close to independent do they have to be before we see something that resembles a binomial distribution?



When do binomial distributions occur?
You are asking, I think, when a [Central Limit Theorem](http://en.wikipedia.org/wiki/Central_limit_theorem) holds. The simplest form of the CLT is that the binomial distributions Binomial(n,p), suitably rescaled, converge to a normal distribution as n goes to infinity. (This binomial case is usually not called the CLT, but goes under the name of the [de Moivre-Laplace theorem](http://en.wikipedia.org/wiki/De_Moivre%E2%80%93Laplace_theorem).)


Now, a Binomial(n,p) random variable is the sum of n Bernoulli(p) random variables. The usual form of the CLT states that if $S\\_n = X\\_1 + ... + X\\_n$, where the $X\\_i$ are independent and identically distributed with mean μ and standard deviation σ, then $(S\\_n - \mu n)/(\sigma \sqrt{n})$ converges in distribution to the standard normal as n → ∞.


If the $X\\_i$ are in fact dependent, see the link provided by Ori Gurel-Gurevich above.


If the $X\\_i$ are independent but *not* identically distributed, then there are two standard conditions for proving that the rescaled distribution of $S\\_n = X\\_1 + ... + X\\_n$ converges to the standard normal: [Lindeberg's condition](http://en.wikipedia.org/wiki/Lindeberg%27s_condition) and [Lyapunov's condition](http://en.wikipedia.org/wiki/Lyapunov_condition). Both are a bit difficult to understand when you first look at them. But the basic idea behind both of them is that if no one of the summands $X\\_i$ is too large (in variance) compared to the others, then the normal distribution still appears.


