When using a Binomial Model for real world situations, it is useful to check if the assumptions of the Binomial model match conditions in the real world. These assumptions are listed below. HOWEVER, it should be noted that models are always wrong, and so the Binomial model may be useful even if the assumptions do not hold -- it depends on the purpose of the model.
Conditions required for Binomial Distribution to be the Number of Successes in N Trials, each of which result in success or failure:
1. N must be FIXED in advance.
2. The probabiity of success p must not vary from trial to trial
3. Different Trials should be independent -- success or failure on one trial should not affect probabilities on other trials..
A You-Tube Video which explains these assumptions of the Binomial Distribution
Example 1: Consider an experiment where we flip coins until we get our first HEAD. Let X be the number of heads. Each trial is Bernoulli, and the number of Heads is the SUM of the Bernoullis, but this is NOT a Binomial. That is because the number of trials N is not fixed in advance. It is easy to see that P(X=1)=p, P(X=2)=(1-p)p, P(X=K)=(1-p)^{K-1} p. This is called a Geometric Distribution.