Poisson probability differs from combinatorial probability, why? I am trying to figure out why the Poisson probability differs from the combinatorial probability.
For example, assume that 10% of people are left handed. What is the probability that a classroom of 30 students contains only right-handed students?
Using combinatorial probability the answer is 0.9^30 = 4.239 %
Using Poisson probability the answer is e^(-0.1*30) = 4.979%
My guess is that the difference is due to the 10% being an observed rate rather than an absolute truth in the Poisson case. What is the explanation?
Edit: Note that if I change the problem to be what is the probability of there being more than 2 left handers in the class then the numbers are dramatically different. The combinatorial probability being 36.1% and the Poisson probability being 57.7%, quite a big difference.
Update: I made a mistake in my calculation. The combinatorial probability for having exactly two left handers is 58.9 %, so it is fairly close to the Poisson estimate.
 A: This is really about the model underlying the maths and which model best fits the scenario you give.
Combinatorial probability is the maths you use when the space of possible outcomes is finite -- which is the case in the example you give (30 students, 2 possibilities for each student).
So there's no question that combinatorial probability is the correct model in this case.
But that just nudges your question into two more precise questions:


*

*why is the Poisson probability so close to the combinatorial probability?

*in what situations would the Poisson probability be the correct one to use?
Here's my answers to the related questions:
1:  Poisson probability is close to combinatorial probability because the Poisson model is the limit of the binomial probability distribution as $N$ (the number of trials) goes to infinity and as the probability $p$ of a success goes to zero.
So you can think of the Binomial distribution as the poor man's Poisson.
2:  The situations in which the Poisson probability is the correct one to use are those where the sample space is continuous.  This can occur when the notion of 'trials', which is a discrete one, seems appropriate even though there is no physical correspondence such as dice rolls, coin tosses, or checking the handedness of a student.
Examples where the Poisson probability is appropriate: traffic probability (accident statistics, chances of cars passing a given junction, etc.), physics (probabilities of events occurring), computer science (probabilities of deadlocking events or other events in operating sysetms), and operations research (e.g. queuing systems, probabilities related to entry / exit / service times in queues).
A: The first calculation you did is the exact probability, under the assumption that we are drawing at random from an infinite pool of students. It is the probability that the number of "successes" is $0$, if we repeat an experiment independently $30$ times, with probability of success each time equal to $0.10$.  
The independence assumption is not quite met, since the population of right-handers is finite, so each each right-hander we find slightly diminishes the probability the next person is right-handed.
The Poisson distribution calculation is an approximation to the binomial. If "$n$" is kind of large, and "$p$" is kind of small, and $np$ is smallish, then the Poisson distribution probabilities, with $\lambda=np$, give a good approximation to the binomial distribution probabilities.
As to when to use the Poisson approximation, there are only rules of thumb. The number $30$ is kind of large, $0.10$ is kind of small, and  $np=3$, in the reasonable range. 
In the old days, the Poisson approximation to the binomial was used more frequently than it is now. The "binomial" probability of exactly $k$ successes is $\dbinom{n}{k}p^k(1-p)^{n-k}$, something unpleasant to compute by hand if $n$ is at all large. Poisson approximation to the rescue, if $np$ is in the modest range. More frequently, normal approximation to the rescue, if $np(1-p)$ is reasonably large.
Nowadays, we have software that computes binomial probabilities for us, so the need for approximations to them has diminished. 
A: The binomial math you worked out is the correct answer. The Poisson is not.  In this example, you are using the Poisson to approximate the answer. But the Binomial is exact. There is nothing wrong with using the Poisson to approximate. Statisticians do this all the time; they use the Normal to approximate the binomial, so on. As long as you know its not exactly correct and why, and how far off you are.
The reason youre allowed to use one to approximate another is because it meets certain conditions that guarantees accuracy within a certain margin: you cannot always use the Poisson to approximate a binomial, but you were able to in this case.
The reason why the Poisson is not the correct one is because you are taking a discrete quantity of successes (left handers) out of a pool of a discrete quantities (people).  The Poisson was not built for this. The Poisson is for taking discrete successes (phone calls, cars at a traffic light, so on) over a continuous variable (such as time).  The binomial was the correct one because it is designed specifically for the model in question.
