MALE SPEAKER: OK, so this morning, Brady sent me an
email, and he said, check this out-- something has happened.
And so, on Sunday, it was the Super Bowl.
I didn't know that myself.
I think Brady-- did you know it was the Super Bowl?
BRADY HARAN: Yes, I stayed up and watched it.
I love the Super Bowl.
MALE SPEAKER: Oh, OK, OK.
Apparently, we have two conferences.
We've got the NFC, National Football Conference.
We've got the AFC, the American Football Conference.
And so every year they have a team from each conference, and
they play the Super Bowl.
OK.
So for the last 14 years-- the previous 14 years-- the team
that represents the NFC have won the coin toss.
Well, it's certainly, certainly a
really unlikely event.
Now unlikely events happen.
So last Sunday, this winning streak of 14 wins that they
had so far was broken.
So last Sunday, it was the AFC, the team that represented
the AFC, was it the Patriots?
The Patriots won the coin toss.
So that's breaking our streak of 14 successive wins.
BRADY HARAN: They then did lose the game, though.
MALE SPEAKER: Oh, that's a shame isn't it?
Never mind.
OK, yeah, enough of this football stuff.
Right, let's do some maths.
Right, so we're going to work out how long it would take to
get that streak of 14 in a row.
14 wins in a row.
Right?
So to make this a bit easier, we're not just going to dive
into the full thing.
We're not going to work out 14 wins in a row.
We'll make this a bit easier.
We're going to look at getting 3 wins in a row.
If we can work this out, if we can understand
it, the rest is easy.
Let's work out how long it would take to
get 3 wins in a row.
OK.
So this is the odds of winning the coin toss,
not the game, no.
I could see Brady doing this.
I thought he was giving me the thumbs-up.
He was like, yeah, good.
Right.
What I want to look at first is-- let's look at getting 3
wins or winning 3 coin tosses in a row.
OK, so first of all, what's the probability of getting
those 3 wins?
This is quite easy.
I think a lot of people can do this themselves.
We're looking for 3 wins in a row.
A win--
a probability of winning the coin toss is 1/2.
So, to win one time, is 1/2.
To win again, you times it by 1/2 again.
And then to win 3 times, you times by 1/2.
OK.
So 1/8 of the time, you could win 3 coin tosses.
What about a streak of 4, a sequence of 4.
Because that means you've lost and then you've won 1.
So it's loss, win, win, win.
Now what's the probability of that?
Again, not too hard to work out.
1/2, 1/2, 1/2, 1/2, 1/16.
1/16 of the time you're going to get a sequence of 4 games
that ends with win, win, win.
Or 4 coin tosses.
So what we're going to work out is the average sequence--
the length of the average sequence that ends
with win, win, win.
What if it was a sequence that was 5?
That means it could be like loss, loss, then ends with
win, win, win.
Or you could have a win, then a loss, win, win, win.
For each of those, the probability is 1/32.
But there's 2 of them, so altogether, if your sequence
is 5 games long, it's 2 out of 32.
So if you want to work out the probability of the average
sequence length, what you do is you take
the sequence lengths--
each situation-- and multiply them by the probability of
getting that situation.
Now this is our waiting time.
This is how long we want to wait for our win, win, win.
It looks like you would have to work out every single
possible situation, every possible probability for that
situation, multiply them all together.
That would give you your average.
That would include all possible situations like win,
win, loss, loss, win, win, win.
Then it would be win, loss, win, loss, win,
loss, win, win, win.
So every single possible situation.
And that does not sound like fun.
So there's a really clever way to do this, and
this is really clever.
Right.
I'll show you how to do this.
I'm going to draw you a little picture for this.
So let's say you lose here and then win.
Or lose and win.
Or lose here and win.
OK, I've drawn you a little picture.
So the probability of losing is 1/2, losing your coin toss.
Probability of winning the coin toss is 1/2.
Let's say we're looking for the time we have to wait to
get these 3 wins in a row.
This call that t.
Now in the first case, I can toss my coin.
And if I lose, I'm back to square one.
I've tossed a coin, and then I'm back-- oh, I have to do it
all over again.
So the length of that sequence--
I've tossed a coin.
Then I'm back to square one.
Back to square one means I'm back to this.
I'm back to waiting t games for my sequence of 3, t.
That's how long I would have to wait if I lose.
And the probability of losing is 1/2.
Times it by the probability, that's what you get.
The other situation is this.
You could win, then lose, and back to square one.
Win and lose probability is 1/4.
And then you're back to square one, where you've tossed it
once, you've tossed the coin twice, and then you're back to
square one.
That means you have to wait t again.
That's how many games you have to wait.
Or the third situation is you win, you win,
and then you lose.
And you're back to square one.
So what's the probability of that happening?
That's 1/8, then 3 coin tosses, back to square one,
means 3 plus t.
Or, and the final situation is you get it straight away--
win, win, win.
And the probability of that is 1/8.
And how long did I have to wait?
3 coin tosses, win, win, win.
And you've got your 3 wins in a row.
Fantastic.
You solve this.
This is what you do.
You solve this formula.
It's not that hard to solve, you just multiply that out,
mess around with it.
I can tell you, if you do you solve it, the
answer is t equals 14.
This is the average length that you would have to wait,
the average number of games you would have to wait to get
3 wins in a row.
Now if you want to do it in general, if you want to do it
for 14, pretty much the same idea.
The same idea.
But in general, you get this.
T is equal to 2 to the power n plus 1, minus 2.
That is the amount of time you would have to wait to get n
wins in a row, n wins.
N is going to be 14 for the Super Bowl.
In the Super Bowl, it was 14.
So if n equals 14, we're going to use this.
And I can tell you the answer.
It was--
32,766.
We would have to play 32,766 games before-- well, that's
the average anyway.
Before you could expect to get 14 wins in a row.
That's the average, I mean maybe it would take longer.
Maybe you'll get it straight away.
We don't know.
But on average, it's going to be about 32,000 games.
Maybe 32,000 years.
BRADY HARAN: It only took 45 Super Bowls for them to get 14
coin tosses in a row, though.
MALE SPEAKER: Yeah it did.
BRADY HARAN: What happened?
MALE SPEAKER: So unlikely events can happen.
Unlikely events like this can happen.
And like I said before, we can toss as many coins as we want.
It's not impossible.
It's just unlikely.
So this isn't a probability.
What I've worked out isn't a probability.
This isn't the probability of winning 14 times in a row.
This is the length of time that we're going to wait, the
number of games we're going to play.
Or the number of coin tosses we're going to do until we get
14 wins in a row.
And it's the average number.
And in the case of the Super Bowl, which happens only once
year, if you want this to happen again, then, I would
expect to wait 32,000 years--
maybe 32,766 years, even.