## Probability Theory - 5 Awesome Things You Didn't Know

Probability theory. Yuck! If you've ever studied it at school or college, then you probably (see what I did there?) hated it as much as I did.

It's boring, taught really poorly, and worst of all, it's boring. Did I mention that it's boring?

Well, the in-depth stuff - which is really mathematical and complicated - is difficult and boring, but basic probability theory isn't what you think it is.

It's actually pretty cool really - and I'm going to prove it to you.

For a while now I've been researching all the little nooks and crannies of the statistics world to create a big picture of what it looks like when you pull it all together.

And this is what the part about Probability Theory looks like:

### Choices, Choices...

And now I'm going to give you a choice...

Ooh, I like choices!

It's up to you...

### Introduction to Probability Theory

Here, I'm going to give you an introduction to probability theory and show you the 5 most awesome things you probably didn't know about, but can make a HUGE difference when planning and preparing your study:

1. 1
Uncertainty
2. 2
Probability Space
3. 3
Theoretical Probability
4. 4
Empirical Probability
5. 5
Law of Large Numbers

And just so you know, this video lesson is actually one of the lessons in The Hive from our exclusive video course Statistics - The Big Picture, which is Open Access to start with (you don't need to register).

If you want to continue learning, you can start from the beginning here:

### Statistics - The Big Picture

#### 1. Uncertainty

Well, to be fair, Uncertainty is likely to be the one thing that you really did know about basic probability theory.

I'd wager that you probably do know. I think you'd get good Odds on it if you wanted to take a chance. In all likelihood...

Yes, as you've guessed I like a good pun.

Statistics is the study of Uncertainty, and we express uncertainty in the form of Likelihood, Odds and Chance.

• Toss a coin - what is the Likelihood that it'll be Heads?
• Throw a couple of dice - what is the Chance that you'll get a pair of sixes?
• What are the Odds of the England football team winning the next World Cup?

Actually, scratch that last question. That's not statistics, it's a mathematical certainty. You've never known the meaning of disappointment until you've supported the England football team.

Oy vey!

#### Pin it for later

Need to save this for later?

Pin it to your favourite board  and you can get back to it when you're ready.

#### 2. Probability Space

Probability Space is the Sample Space where Events happen.

Huh? What does that mean?

To explain this, let me take you away from statistics and probability for a moment and let's do a little thought experiment.

Close your eyes and picture a cardboard box.

This is your Probability Space, and it's where you are going to plan and do your experiment.

Everything you need in your experiment, I want you to visualise placing it inside the box.

You're doing a study about female breast cancer, so visualise yourself placing 'Breast Cancer' inside the box. Also, visualise putting 'Gender: Female' into the box.

Probability Theory - 5 Awesome Things You Didn't Know (video) #probability #probabilitytheory @chi2innovations @eelrekab

In the same way as all the things you need go inside the box, all the things you don't need go outside the box.

So visualise placing 'All Other Cancers' and 'Gender: Male' outside the box.

You do this for every variable you need, and don't need.

This may sound theoretical, and it is, but Probability Space is also where you do your actual experiment - it is a Universe in which you construct a theoretical experiment, run a real experiment and then compare what you expected with what you actually got.

It's actually a very powerful way to plan your experiment, because you have a visualisation of everything that goes in, everything that goes out, and everything that happens in your experiment. You only need to close your eyes to remember all the decisions about your study that you made (although I strongly suggest that you write them down too!).

#### 3. Theoretical Probability

The Theoretical Probability is all the things that you expect to happen in your experiment - inside your Probability Space (the box in your mind).

Toss a coin 10 times - what do you expect the outcome to be?

• 5 Heads and 5 Tails

That's the mathematical reasoning. That's the Theoretical Probability.

#### 4. Empirical Probability

The Empirical Probability is all the things that actually happen in your experiment.

Toss a coin 10 times - what will the outcome actually be?

• 7 Heads and 3 Tails
• 2 Heads and 8 Tails
• 6 Heads and 4 Tails
• ...

You don't know what you're going to get, and this is where the Empirical Probability differs from the Theoretical Probability.

#### 5. Law of Large Numbers

The Law of Large Numbers is where all the magic happens.

This tells us that in the long run, if we do enough experiments, the Empirical Probability will be the same as the Theoretical Probability.

This is absolutely HUGE because it tells us that we can do small scale experiments and then extrapolate the results to the larger scale - as long as we have enough samples in the experiment and we run the experiment lots of times and keep getting (mostly) the same result.

This is why we can run a breast cancer study with only 200 patients and then make treatment decisions based on the results that effect the lives of a million other women around the globe.

Just imagine if the Law of Large Numbers wasn't true.

We would live in a world where randomness ruled, and nothing would emerge from the randomness. 5 Heads and 5 Tails would not be the most likely outcome of repeatedly tossing a coin 10 times - all possible outcomes would be equally likely.

We couldn't do small scale experiments. We couldn't learn. We couldn't teach. We couldn't improve.

At best we would still be cave men, and at worst we wouldn't exist at all.

The biggest evolutionary advantage we have (apart from opposable thumbs) is our ability to think and work stuff out. Without the Law of Large Numbers, we wouldn't have that and we would probably not have evolved beyond the apes.

The Law of Large Numbers is just AWESOME!!!

### Basic Probability Theory - A Summary

When it all comes down to it, basic probability theory is waayyyy more awesome than we all thought it was.

Basic probability theory is more than a theoretical thought experiment - it's a mind box where we plan and do our experiments. It's where we compare what we expect to happen with what actually happened when we ran our experiment.

And best of all, we're safe in the knowledge that whatever we do (if we do it right) the results will apply to wider society.

I don't know about you, but I think Probability Theory is one of the coolest kids on the block...

I appreciate that Probability Theory isn't the sexiest beast on the planet, so if you've got this far without running away, well done!

Give yourself a boost by clicking our 'Awesome Button'!

### Statistics - The Big Picture

I hope you enjoyed this video lesson.

It's actually one of the lessons in The Hive from our exclusive video course Statistics - The Big Picture, where I take you through all the nooks and crannies of the statistics world.

This course is Open Access to start with (you don't need to register), and after you've gone so far through you'll need a Free Plan to continue.

As part of the course you'll get an Ultra-HD pdf of Statistics - The Big Picture to download and keep, and I'll share with you where you can also get your own poster to pin on your wall.

If you want to continue learning, you can start from the beginning here:

### Statistics - The Big Picture

I look forward to seeing you on the inside!