derive binomial distribution for a classroom

Question

I am supposed to give a 20-min presentation to a group of kids aged around 14-15, so they have very limited knowledge about the topic in advance -- at the same time I want the talk to be something they can relate to.

I am thinking of presenting the derivation of the binomial distribution since it is a concept that is relatively easy to grasp (think toss-coin etc..) and can be done in 20 min.

However, my concerns are

1. How do I easily introduce/derive the binomial coefficient?
2. Once the distribution has been derived, do you have any suggestions for what we can do in the classroom do illustrate it that involves the students and doesn't take more than 2-3 minutes?

I'd be happy to hear your ideas and suggestions.

Answer 1

I would define $\binom{n}k$ to be the number of different $k$-subsets of an $n$-set without initially worrying about how to compute it. This is all you need to explain why the probability of $k$ successes in $n$ trials is $\binom{n}kp^k(1-p)^{n-k}$.

Then you can return to the problem of computing $\binom{n}k$. I’d explain why any one of those $\binom{n}k$ $k$-subsets can be lined up in $k!$ different orders, so the number of ways to pick out $k$ elements and line them up must be $\binom{n}kk!$. On the other hand, the same reasoning that gave us the $k!$ shows that there must be $n(n-1)\ldots(n-k+1)$ ways. Observing that we can write this more compactly as $\frac{n!}{(n-k)!}$ we solve for $\binom{n}k$ to find that it must be $\frac{n!}{k!(n-k)!}$. Alternatively, you can just leave it in what is often a more useful form,

$$\binom{n}k=\frac{n^{\underline k}}{k!}=\frac{n(n-1)\ldots(n-k+1)}{k!}\;,$$

without rewriting the falling factorial as a quotient of factorials.

For an illustration you might have each of the kids toss a coin a dozen times or so, recording the results. If there are enough kids, you can compare the outcome with the theoretical binomial distribution for $n=12$ and $p=\frac12$. If there aren’t enough kids, you can have each of them repeat the experiment.

Answer 2

I like to start the binomial coefficients in combinatorial interpretation (# of way to choose $k \ge 0$ from $n\ge k$), it only requires the Multiplication Principle, which is straight-forward. (You can also throw in elements of Pascal's triangle, which will make a nice puzzle for kids who like independent thinking).

Then apply this to coin tossing.

Answer 3

As I recall, derivation of the binomial coefficient begins with an explanation of the use of factorials to count permutations.

That is: "$n$ distinct objects can be arranged in a line by choosing one of $n$ as the first place, one of the remaining $n-1$ for the second place, and so on.   Thus there are $n\cdot(n-1)\cdot(n-2)\cdots 2\cdot 1$ distinct ways to do so.   We represent this as $n!$" or something like that.

Then you explain how the count of arrangements of two groups, of $k$ and $n-k$ items distinguishable only by, type is $\frac{n!}{k!(n-k)!}$.   (Red and blue M&Ms are a good example.)

Next show the expansion of the binomial $(p+q)^n$ ("a binomial is a polynomial which is the sum of two terms").   Which is: $(p+q)^n = q^n + n p q^{n-1} + \cdots+\frac{n!}{k!\,(n-k)!}p^kq^{n-k}+\cdots+ n p^{n-1}q+p^n$.   "To determine the coefficient of the term $p^k q^{n-k}$ we count how many ways to select $k$ of the $p$ terms and $(n-k)$ of the $q$ terms from $(p+q)(p+q)\cdots(p+q)$" or such.   Usually begin via demonstrating the expansion of $(p+q)^2, (p+q)^3$ and then generalising the process.   "This is why it is named the binomial coefficient."

Finally, explain how this relates to the Probability Mass Function of the count of successes in $n$ Bernoulli trials, each having a success rate $p$ (and failure rate $q=(1-p)$), and thus why this is called a Binomial Distribution.

Well, that's basically it.   Use your own words.