Did computers render useless the teaching of approximating the Binomial with Poisson and Normal distribution?

Question

When considering a binomial distribution with large $n$, it is (was?) usefull to use the Poisson or Normal distribution instead. One of the reason being the difficuty to compute the binomial coefficient for large $n$. Computers have since grown in performance and algorithms in efficiency.

Is it still worth teaching students about approximating $B(n;p)$ with $Po(np)$ for small values of $np$ (or small values of $n(1-p)$) and with $N(np;np(1-p))$ otherwise?

An ideal answer would include reference (personal or theoretical) to what is actually done in practice and to the applications of such approximations in undergraduate or graduate courses.

Answer 1

They are indeed not so useful as calculational tools, but in terms of the theory they are important. The approximation of binomial by Poisson motivates many applications of the Poisson distribution. The approximation of binomial by normal leads to the Central Limit Theorem.

EDIT: A classic example of a practical application of Poisson distribution (from von Bortkiewicz in 1898) is the number of deaths in a year in the Prussian army due to accidental horse-kicks. We may imagine that there are a large number $n$ of Prussian soldiers, each of whom, independent of the others, has a small probability $p$ of being accidentally kicked to death by a horse in a given year. The total number of such deaths would then be binomial with parameters $n$ and $p$. Since $n$ is large and $p$ small, the result should be well-approximated by a Poisson distribution, and indeed this is what von Bortkiewicz found. Actually the probability will vary from one type of soldier to another, but this does not affect the result.

Answer 2

The Central Limit Theorem, which is the topic of your question, is still of very practical use.

Most applications of $z$-tests and $t$-tests and several other hypothesis testing methods rely on the Central Limit Theorem for their validity.

Take election polling, for example. Suppose you have a voting populace of zillions of people who choose from amongst two candidates $A$ and $B$, where the the proportion of $A$ voters to $B$ voters is $p$ to $1-p$, and where $p$ is not too close to either $0$ or $1$. A pollster (in the ideal setup) randomly chooses some number of voters, say 1000 of them. This sample can be thought of as a sample taken from the distribution $B(1000,p)$. The pollster then wants to apply hypothesis testing methods, i.e. the $z$-test or $t$-test, in order to determine confidence intervals for the mean, i.e. confidence intervals for the pollsters prediction of the election outcome. What justifies applying those tests is that $B(1000,p)$ is very, very close to a normal distribution.

The same justification applies every time a medical trial samples amongst patients who do or do not respond to a medication in a proportion $p$ to $1-p$.

It also applies to a particle physics experiment when sampling particles that do or do not undergo decay in a proportion $p$ to $1-p$.