Lets say you got two people. You got Person A Person B, right, they're flipping coins.

Let's say one of them is flipping coins and waiting for Heads-Heads to turn up

right, so they're going to make a sequence of coin flips, and they're waiting for Head-Heads.

Person B, he's doing it and he's waiting for Head-Tails.

So one of them is waiting for Head-Heads, and one of them is waiting for Head-Tails.

Now, what's the probability that you're gonna get Heads-Heads?

Now even Brady would probably know this. What's the probability you get Heads-Heads?

Brady: Is it one in four?

Yeah yeah exactly. So it's one in four, and Heads-Tails it's the same, right? It's one in four.

Even though they have equal probability I claim your average waiting time for Heads-Heads

is going to be longer than for Head-Tails.

Let's do it, let's do the experiment. Let's not just write out mathematical formula.

Let's do it let's go.

I'm gonna flip this coin. Now this is, we're gonna do an average. I wanna do a long sequence

and we're gonna see what the average is.

So if I flip this coin, let's say 50 times, is a nice long number.

Then we'll be able to take an average of how long we have to wait for Heads-Heads or Heads-Tails.

Right, so we do this if it... actually this is gonna take me ages, isn't it? So instead of flipping this one coin 50 times

what i can do is I can flip 50 coins all at once alright

which I've just done there, OK?

The First coin is a Tails alright. Second coin is, without looking, it's a Heads.

Alright, next coin is Tails, I'm gonna a sequence like that.

Not even Looking.

We've run out of space we better do another.

Let's make them random.

Right, so I poured them out, I tried not to look at them, so it should be a sequence of random coin flips.

I better write out so we can see what they are a bit better.

Tail...Head...Tail...Head...Tail...and a Head and a Tail and a Tail...Tail...Head...Tail

Alright, we're gonna look at this sequence and we're gonna play the game

where tossing a sequence of coins.

And we're waiting to see how many, how long we have to wait for Head-Tail.

If i start here I'm looking for a Head-Tail, Oh, there it is. Right.

Alright that's my first Head-Tail there. That was a waiting time of three

and then i got Head-Tail, then I start agina OK. I do a new game

I'm gonna look for Head-Tail, and I had to wait one two three four, and there it is.

Now i'm gonna do it again, oh look this one is straight away look Head-Tail.

Now I'm gonna start again. I'm looking for Head-Tail. Ooh this is a long wait here.

And then there it is Head-Tail.

And Head-Tail there...there...there it is. Oh and there's one right at the end as well.

Right, what was the average waiting time. Let's have a look at the waiting time.

So that's about an average wait of 4.5 .

Lets look at the Heads-Heads though. You would have to wait, ooh how long? Ooh this is a long waiting time here.

It happens all the way over here.

There Heads-Heads, which was a long wait. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15.

So you've player a game and it ends when you get Head-Heads. Great, done.

So now start a new game, a completely different game form this point on.

So we'll just start again, oh, but this time notice there the Heads-Heads comes up really quickly. Look at that, it's just straight away.

And now we start a new game OK. So we had a success, we'll start a new game and we have to wait 1, 2, 3

let's look for our average waiting time for Heads-Heads, OK.

Which is 7, right. And yes, oh look, yes.You have a longer average waiting time for Heads-Heads than you do for Heads-Tails

even though they have an equal probability, they're both one quarter. Why is this?

You can kind of see what that reason is from this, this sequence.

Because if you noticed when we were doing it, when you play Heads-Tails

you get Heads-Tails, then you start a new game and you wait for Heads-Tails, and you start a new game.

Heads-Heads was slightly different, when we did it with Heads-Heads look what happened here.

when you have something like this: Heads-Heads-Heads-Heads we have this overlap.

And this overlap isn't counted, we were playing it until we got Heads-Heads

and then the game stopped. And then we play a new game from that point.

So here we've got a Heads-Heads, that didn't get counted as a Heads-Heads.

So if you look at how many Heads-Heads we've got it should about an equal number to Heads-Tails

but the overlaps don't get counted. Let's just check.

How many Heads-Tails did we have? Actually we know we got 11, i know for a fact we got 11.

How many Heads-Heads did we have, including the overlaps?

One here, we have two, we have three.

So if you actually count up the Heads-Heads we had 9 of them, but only 6 were included in our waiting time average.

So this sequence, we were looking at what would happen for consecutive values

so Head-Heads, Heads-Tails. The same sort of thing would happen with Tails-Heads and Tails-Tails.

We're look at consecutive values, in fact the reason I mentioned this is because

if you remember there was that prime news where they were looking at primes, and the ending of primes for consecutive primes.

This is what they were looking for, they thought if primes were random like coin flips

I'm going to find this effect.

What turned out to be the case is that they didn't find this effect because the primes weren't being random like coins.

If you did this forever, if you had a sequence that went on to infinity the average waiting time then is

I'll show what that is. It's called expectation or expected waiting time.

The expected waiting time for Head-Tails is equaled to 4

Which kind of make sense when it's a quarter probability. That does make sence.

And then the expected waiting time for Heads-Heads is longer, and it's 6.

So we were close but just a little off. And if you wanted the expected waiting time for Tails-Tails, well that's 6 as well.

That's just the same idea.

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and then the free trial for 30 days. Our thanks to Audible for supporting this episode of Numberphile.

Maths works, maths just works, it's wonderful that way how maths just works out

Brady: Every time.

Every time just how you want it to work out. Especially with the power of editing.

Brady: What happened?

What happened, we did out first take and we got the opposite result from what we wanted.

Brady: It's done you

48 over 11, which is the opposite of what I said.

And we have to film the whole thing again, right?