Hello Unenlightened Originally Posted by

**Unenlightened** Right, but what happens when the exam question has 10 coin tosses?

They can't write out all the combinations then...

Have you thought about playing around with Pascal's Triangle? Lots of information on the web. I just Googled Pascal's Triangle.

Each row represents the distribution of heads and tails for a given number of tosses. For example on row 4 (counting the top '1' as row 0):$\displaystyle 1\quad4\quad6\quad4\quad1$

the sum is$\displaystyle 1+4+6+4+1 = 16$

This sum represents the total number of different ways $\displaystyle (2^4)$ in which the $\displaystyle 4$ coins can land.

Each individual number in the row gives the number of ways of getting a particular number of heads. Thus$\displaystyle 1$ represents the $\displaystyle 1$ way of getting $\displaystyle 0$ heads and $\displaystyle 4$ tails: TTTT

$\displaystyle 4$ represents the $\displaystyle 4$ ways of getting $\displaystyle 1$ head and $\displaystyle 3$ tails: HTTT, THTT, TTHT, TTTH

... and so on

The probabilities associated with each of these numbers are, of course, the numbers divided by $\displaystyle 16$.

By playing around with the first few rows of Pascal's triangle, then, it should be fairly easy to develop a feeling for the different combinations and probabilities when the numbers are small. It's then a simple matter of extending the triangle down to row 10 - or indeed, of developing the formula$\displaystyle \binom{n}{r} = \frac{n!}{r!(n-r)!}$

and, if you're feeling ambitious, the Binomial probability formula$\displaystyle p(r) = \binom{n}{r}p^r(1-p)^{n-r}$

which is particularly simple if $\displaystyle p=\tfrac12$:

$\displaystyle p(r)=\binom{n}{r}\left(\frac12\right)^n$

As an alternative, if computers are available, have you thought about using a random number generator in a spreadsheet to simulate the throws of the coins? Excel, for example, has an Analysis Tool Pack that is most useful. I found (before I retired!) this a very successful method of teaching the early concepts of probability. It's very easy to simulate a large number of experiments - begin with just two coins, for example, 'tossed' 1000 times, and look at the distribution. You'll get approximately 250, 500, 250, but it won't be exact, and can then lead into a discussion of experimental versus theoretical probability. I found that getting the computer to do all the hard work allows the students to develop a solid understanding of the basic concepts by handling realistically large amounts of data.

Grandad