# Heads, Tails And Beans |S|=1

Mathematics is not everyone’s cup of tea – fair enough. The Monday morning ‘double maffs’ articles by our resident mathematician, OldTrout, can present some GP’ers with a feeling of bewilderment. Myself included.

I’m no mathematician. My mathematical knowledge is small but I like to try. There’s a personal reward in solving a mathematical problem, however simple.

Most of us have studied (at school) a little algebra and perhaps calculus. The branch of mathematics that OldTrout studies and writes about is called combinatorics.

As the name suggests it looks at the patterns (combinations and permutations) of not only numbers but anything you like and applies mathematical functions to count new information. A simple example: In an eight horse race how many ways are there for all of the horses to finish in the 1st, 2nd and 3rd positions?

In the following articles I’ll introduce you to this fascinating subject with the examples of games: playing cards, dice, lotteries and coin tosses. Essentially, we’ll learn how to count smarter.

Impossible Or Improbable?

I think that we have all said at some point, “this is bloody impossible!”. Of course, what we really mean is that this is very difficult. If I threw a tennis ball into the air, say a hundred times, and expected it to stay there (apart from me being an imbecile) it would indeed be reasonable for me to say, “this is bloody impossible!”.

Possibility is either true or false. Throwing a ball into the air and expecting it to defy gravity is not possible – it’s false. Threading a sewing needle might be difficult but it is possible – it’s true. We say that something is impossible if it has 0% chance of happening and possible if it has some chance of happening. Something can’t be slightly possible or slightly impossible.

Flipping a coin to determine the outcome of two choices is a game that we all learn from an early age. The coin has a 100% chance of being either heads or tails (we’ll ignore the coin landing on its edge!). The players have a 50% chance, or probability, of winning.

Therefore, it’s only possible for one of the players to win but probable that either of them will.

We can think of possibility as an on/off switch – a light can be on or off. Probability is like a dimmer switch – somewhere in between the two states.

Is it possible to win a lottery? Of course, 100%. Is it probable that I’ll win a lottery? Err…sorry, no. The current UK lottery has you choosing 6 numbers between 1 and 59 inclusive. That’s 45,057,474 (over forty-five million) unordered 6 digit tickets for the big win.

A 0.0000022% probability of owing a super yacht per ticket. Possible but highly improbable.

Going Forward

I’m going to try and keep these articles short. Reading a long and technical document does not help anyone. Bite sizes of ideas and information are the best. In the next part I’ll start to try and explain some of the formal mathematical notation of combinatorics.

A special thank you to OldTrout for the help with LaTeX, MathJax and dice – I tend to make unnecessary boxes to think out of. Below, hopefully, is a formula that describes all possible lottery tickets:

$B_k = \{T \in S:|T \cap T_0|=k\} \subseteq S$

…and by the end of this series we should be able to understand it! Oh, the answer to the horse race question is 336.

(Note: You might not be able to see the mathematical formula properly in your web-browser if you have a script blocking plug-in or Java-script is disabled.)