poisson distribution VS binomial distribution I have a really difficult time when I read questions deciding if I have a poisson distribution or binomial distribution. So I am curious are there any key words or key details I can look for to decide if I have a poisson distribution or binomial distribution? I did try to read through previous examples of this. Any help or explanation would be greatly appreciated :). 
 A: If you mentioned any particular instances in which you have difficulty deciding between the two, maybe it would be possible to write a more informative answer.
The binomial distribution is the probability distribution of the number of successes in a fixed number of independent trials with the same probability of success on each trial.  "Fixed" means non-random, i.e. the probability that the number of trials is some specified number is $1$.
The Poisson distribution assigns positive probability to each non-negative integers $0,1,2,3,\ldots$, whereas the binomial distribution assigns positive probability to each of $0,1,2,\ldots,n$, where $n$ is the number of trials, and probability zero to everything else including integers bigger than $n$.
As the number of trials grows while the probability of success on each trial shrinks in such a way that their product, which is the expected number of successes, remains equal to some positive number, then the binomial distribution approaches the Poisson distribution.  Suppose one asks what is the probability that the number of successes is $5$ if there is a one-in-a-million chance of success on each trial and there are $3.8$ million trials.  Then the expected number of successes is $3.8$, and the probabliity that the number of successes is $5$ is
$$
\binom{3\,800\,000} 5 \left(\frac 1 {1\,000\,000} \right)^5 \left( 1 - \frac 1 {1\,000\,000} \right)^{3\,800\,000 - 5} \approx \frac{3.8^5 e^{-3.8}}{5!}.
$$
The number on the left is exact and is what is given by the binomial distribution, and computing it exactly is expensive.  The number on the right is what is given by the Poisson distribution, and probably differs from the number on the left only after a large number of digits after the decimal point, and can be computed quickly.
