# 0. Title Page

Combinatorics for Cats

Probability for Pussies

Functions for Felix

Featuring assorted other fluffy creatures.

(c) Vitenka 2006

# 1. Intro

Let's start right at the beginning.

In the beginning

The earth was without form.

And void.

Then someone drew a cat.

You can tell it's a cat, because the author says that it is.

That's an assertion. You have to take them on trust, until you've seen them do enough useful things that you're convinced.

# 2. What is probability for?

Curiously, my curious cat asks an important question.

It does lots of stuff.

Mainly it lets you answer the question "What're the chances?"

Which is important for lots of stuff.

Such as making new silicon chips, or planning the best place for a housing estate.

Or gambling. Probability was invented for gambling.

# 3. Noddy Gambling example

It's time to play the tupenny stalls at the church fete. Pretend that you're enjoying yourself.

One stall is offering to pay out to anyone who can roll a double six.

Another pays the same amount to whoever can pull an ace out of a pack of cards.

Which game would it be better for you to play?

Why?

# 4. The experimental method.

Well, this one is easy. Grab some dice. Grab a pack of cards.

Which one pays out more often?

... why are we waiting? Why-eye are we waiting...

... come on. Roll them dice. Draw a card then shuffle and do it again. Come on! ...

Ok. So it looks like the cards pay out about three times as often as the dice do.

But sometimes the cards don't pay out for a while. And sometimes the dice pay out two or even three times in a row.

Probability can't tell you what will happen.

It just says how likely it is to happen.

And it mainly does this by saying what happens in the long run.

# 5. Multiple Equally likely outcomes

Right. Time for one of those assertions.

The chance of 'n' equally likely outcomes is:

- Pronounced 'one over enn'

Oh! Sorry. We need a scale.

Some people like percentages.

Those people are weird.

Some people like ratios.

Those people lose all their money down the track.

# 6. Scale

We're mathematicians.

Well, we're pretending to be.

We're going to have a scale of one down to zero.

One is the top. Something with a probability of one is certain to happen. (Same as 100%. And there's no way to write that in betting language.)

Something with a probability of zero can never happen. (0%) And again...

A probability of ½ is 50%, or 1:1. It happens half the time.

# 7. Sensibleness of Multiple Equally likely outcomes

Right. Lets convince ourselves that our one-over-n rule is sensible.

I hid a cat in one of these boxes.

There's no way to tell which one I hid the cat in, each is equally likely.

If you open one (without picking them up, or shaking them, or listening for mewing or anything) - what are the chances that you will get your face clawed off by an angry escaping cat?

Well - it's one in three. 3:1 or 33.33333...% - or 1/3.

Did you pick right?
Pick up your face and we'll move on.

# 8. Conviction.

Let's convince ourselves.

First of all, all the probabilities must add up to one.

If we open all of the boxes, we certainly opened the one with the cat in.

(That's a rule - but it's also kind of obvious. We have to open one of the boxes, and a certainty has a probability of 1. Also, think what happens if you only have one box...)

But we also said that each box was equally likely to be chosen.

So we've got two things known.

1. The total is one.
2. Each chance is the same.

Oh, and one thing more.

3. There are n (and today n=3) boxes.

# 9. Notation and Algebra

Let's write this down. We like symbols in maths - so.

We'll write the probability that the cat is to be found in box one as:

And so on.

Then we can write:

And

A little bit of maths sorts this out for us.

# 10. Back to the fete. Cards.

So. There are 52 cards in a pack.

So the chance of choosing any particular one of them

must be 1/52.

There are four aces. So the chances of drawing one should be:

Roughly one time in thirteen you will draw an ace.

# 11. Percent-age

Ok, I'm going to abandon percentages and horsey rules entirely from now on.

Percentages. They range from 100% meaning completely true, to 0% completely false, nice and linear (50% is half true)

That's easy then. Multiply the 0-1 number by 100, and it's a percentage.

Pack your bags, percentage lovers - you're outta here.

Right. Ratios are a bit more complicated, and because I hate them I'm not even going to justify myself.

Expressed as a probability, a:b (against) = b / (a+b)
( 3:1 on is the same as 1:3 against = 3/4 )

(e.g. 2:1 = 1/3, 5:4 = 4/9 )

Note that while ratios are often pronounced as 'to' (e.g. 1:2 is 'one to two') - probabilities are often pronounced 'in'. (e.g. 2/13 is 'two in thirteen') This is because you can just multiply up. If you do something with a probability of success of 2/13 thirteen times, you'll expect to see it happen two of those times. So 'two successes in thirteen tries' - or just 'a two in thirteen chance'.

# 12. I slipped in the rule ( P(A=a u A=b) = P(A=a) + P(A=b) <=> P(A=a n A=b) = 0 )

Notice my cheating, a couple of pages back? I used a rule without introducing it.

Maths tends to be like that - to get anything done you have to mix a whole bunch of stuff together. Still, it's best to at least try and keep your feet on the ground.

So, the unstated rule was that you can add a bunch of probabilities together.

And you can - sometimes. It isn't always safe. For example; flipping a coin. By our rules, the chances of a head are 1/2. But if we flip the coin twice - the chance isn't 1. It's possible to flip two tails in a row - so we're surely not certain to get a head.

It is safe when:

• You're talking about ENTIRELY SEPARATE outcomes. (The chance of BOTH of them being true has to be zero)
• You're talking about THE SAME TEST. (So we CAN say that the chance of getting either a head or a tails on a coin flip is 1 - but we can't say that if we look for a head on one coin flip and a tails on the next coin flip.)

I'm not gonna prove that. You'll have to take it on trust. Fiddle about with some coins and stuff to convince yourself.

# 13. (that rule, continued)

Ok, fine. Let's do that rule.

The full rule is:

First part - it has to be the same test. The coins example is a proof that the opposite isn't true - but it doesn't prove that it IS true for a 'same test'.

For that we need another example.

One cat. One box. The possible outcomes are 'the cat is in the box' (probability 1) and 'there are flying monkeys on the moons of jupiter' (probability zero) and a bunch of other impossible probabilities.

We already said that the total probability has to be one. So there's an example - it is at least sometimes true.

Now for the second part.

If the two events 'a' and 'b' overlap, then it's not true any more. For example - let's have 'a' be "I rolled 4 or less" and 'b' be "I rolled 3 or more".

Now, if I roll a 3 or a 4, both a and b are true. So if we add them up then we have counted three and four twice. Which would add up to more than one. Since the probability can never be more than one, we know that's wrong.

On the other hand, we can add up EVERY possible 'a' and 'b' that way - and if ANY of them came to more than one, we'd know we counted something more than once.

So the only way it can be safe to add them together is if they don't overlap.

. . .

Ok, that's a hand waving 'trust me' not a real explanation. It's all you're getting.