So, what I'm going to ask you to do, is to pick a sequence of three coin tosses.

So, you know, heads-tails-heads, or tails-tails-tails.

Brady: "Tails-tails-heads."

Okay. So, Brady's picking tails-tails-heads.

And I

will pick heads-tails-tails.

And the game is: we're gonna toss this coin and see who gets that first.

So, let's see who wins. We'll do a best of 5.

Okay? Something like that.

Okay, so,

let's try game Number 1. All right, here we go.

So, number 1...

All right, let's write down this sequence.

Heads.

Okay, that's pretty good actually; that's good for me.

Yeah, it's good for me, not good for you.

All right, heads again.

Heads again.

Ooh, look at this.

We might have to do a different video about how many heads we can get in a row.

Oh look, tails. Alright

Oooh, tails.

That is a win for me. Excellent.

A James win, brilliant.

Alright, well, best of 5?

Alright, Brady's got a tail. Come on.

Brady: "Ah, now, this is good for me."

Yes! That's a tail.

Brady: "Yes!"

- And...

It's a tail!

Oh no you're okay though

You're still, you're still in the lead

Yeah, happy with that aren't you?

Alright, oooh

Brady: "That's good for me."

Tails, tails

Brady: "In fact, I'm definitely gonna win this next one"

Soon as the sequence breaks, yeah,

you're right.

Well observed, Brady.

Brady does win that, he had to win that after all those tails coming up. Okay,

Brady: "So, It is bad to choose any two consecutives as you start."

It is bad to choose.

Brady!

Brady: "I realize what I did wrong."

Ahhh cmon cmon cmon you can still win this

Have you given up?

Brady: "I've realized my mistake."

You've realized what a mistake

You're less likely to win than I am.

Brady: "So, what would you have done if I had not been stupid and not given consecutives as my..."

Whatever sequence that you pick

I can pick a sequence that is more likely to appear before yours.

Brady: "What if I pick the most likely of them all?"

Exactly, what if you pick the most likely of them all?

It goes round in a circle like rock, paper, scissors

Let's say that Brady picks whatever sequence he does pick

I'm going to show you what sequence I should pick so that I can beat him.

Right, or at least with a better chance of beating him.

I'm going to show you the cycle of winning. Let's do that.

Right. So let's say Brady picks heads-heads-heads, right? Three heads in a row.

Then what I should pick is tail-heads-heads.

And tail-heads-heads will beat heads-heads-heads.

If Brady had picked tails-heads-heads,

then I should choose tails-tails-heads.

Because that is going to beat Brady.

If Brady had picked tails-tails-heads,

then I should pick heads-tails-tails, that will beat Brady.

Incidentally, if he had picked, instead, tails-heads-tails,

you pick this one here, tails-tails-heads. So it's not quite a circle, this.

It has some spikes coming off the side of it.

If Brady had picked tails-tails-tails, you want to choose heads-tails-tails.

If Brady had picked heads-tails-tails,

then the best choice is heads-heads-tails.

If you had picked heads-tails-heads, I would again pick heads-heads-tails.

If you had chosen heads-heads-tails, Brady, I would have picked tails-heads-heads.

So we have this little circle, but you can see the spikes coming off it as well.

So to read this, Brady's choice would be here, at the pointy end of the arrow.

My choice, the better choice, would be here, so the arrow going towards Brady's choice there.

There is a way to help you to remember this cycle.

The way you do it is, when your opponent picks a sequence, let's say, tails-heads-heads,

what you should do is, in your mind, copy the middle one, so make a copy of that coin,

so I've got another heads there, I'm gonna put it at the front, and flip it over.

So it would become a tails. And then my choice would be tails-tails-heads.

That's the winning choice.

That little way of remembering it is better than just remembering that cycle.

And that will work. That's your best choice to go for.

I should show you the probabilities for each choice.

If Brady picked heads-heads-heads, I would pick tails-heads-heads.

And the chance of beating you is actually pretty good. It's seven eighths. I can tell you that's round about 87.5 percent.

So it's really likely. I'm really likely to win the game.

That's why I did best of five, though, just in case probability let me down.

This probability here, if I wanted to work tails-heads-heads beating heads-heads-tails,

that happens with a chance of three quarters, so it's about 75 percent,

really big probability.

If you wanted to do this one here, heads-heads-tails beating heads-tails-heads,

that happens with a chance of two thirds, so that's, what's that, about 67 percent.

And then, actually the others are similar.

This probability here is two thirds, this one here is seven eighths, this one here is three quarters.

There's a symmetry in this.

So you are far more likely to win.

Excellent. Try it out on your enemies, right. So you can beat them.

This is insane, because you think, well, surely there's a best choice.

There's a best choice, and then, all the others are worse than that.

And you can't beat the best choice.

It works in such a strange way that it makes this cycle of probabilities instead.

Just to show you where these come from. These are some of the easy ones, okay?

I'm gonna show you where some of these come from. Well, look, this is an easy one, look at this.

I said this was seven eighths. Yeah, that's actually an easy one to spot,

because, well, you could get heads-heads-heads straightaway, which happens one eighth of the time.

But if you don't get heads-heads-heads straightaway, if you've got a sequence,

and somewhere in the sequence it's heads-heads-heads, let's say this is the first appearance,

then it has to be preceded by a tail.

If it's not, if it was preceded by a head, then it wouldn't be the first heads-heads-heads in the sequence.

It has to be preceded by a tail.

Which means it has to have tails-heads-heads coming up before it

unless

you get the three heads in a row straightaway.

So, yeah, you're going to lose if you pick heads-heads-heads, most of the time.

Yeah, so we go back to our game, what Brady chose. He chose tails-tails-heads, it wasn't the best choice.

Brady, if you picked tails-tails-heads, I used my little algorithm, I know what I'm supposed to choose,

heads-tails-tails, and I can beat you, appear before yours does, with a chance of 75 percent.

Which is actually the second best thing there.

Really bad choice, Brady. Sorry about that.

Well I might not be making the best coin choices, but I have been learning a lot of new mathematical tricks

at TheGreatCoursesPlus. This is a great online resource for anyone who wants to learn anything.

Become smarter about things from cookery to quantum mechanics.

I think numberphile fans, in particular, might be interested by some of the mathematical offerings.

They're really extensive, and right up the alley of the sort of people who watch these videos.

There's lots about games, puzzles, probability. I've really been enjoying, just today, some of the videos about probability.

I wish I'd watched them before recording that video with James.

If you'd like to find out more, go to TheGreatCoursesPlus dot com slash numberphile.

Have a look what's there, and if you like it you can actually sign up for a one-month free trial.

That's one month's access to 7000 plus videos, all taught by leading experts in these fields.

That address again, TheGreatCoursesPlus.com/numberphile.

Give it a look. If you like these videos here on numberphile, I think you might really like what you find there.

And cheers to the GreatCoursesPlus for supporting this video.

This game is called Penney's Game, or penny ante, which I think is a kind of a pun.

Actually, the strange thing is, Penney doesn't refer to coins and pennies, and flipping coins,

It's actually the name of guy who came up with the game. He was called Penney.

It's kind of one of those situations when you have a baker called Mr. Baker,

Or a mathematician called Dr. Sexy or something like that.