Alex: We have got a brilliant and very simple probability puzzle about a ball and a bag,
well maybe there's more than one ball.
But before that I want to tell you about this puzzle, and the interesting thing about this puzzle, which very few other puzzles
you can say, is that I know
exactly the time it was invented.
It was invented by Lewis Carroll as he lay in bed on the 8th of September, 1887.
How do I know that? I know that because
Lewis Carroll wrote a book of
problems that he called his pillow problems, and he called them his pillow problems because;
I don't know about you, when you're trying to go to bed,
but Lewis Carroll
who I should really say maybe he doesn't even need to be introduced
he's the guy who wrote Alice's Adventures in Wonderland.
He said that: "when you are lying in bed, sometimes you are victim to
*skeptical thoughts*, which seem for the moment to uproot the firmest faith or
*blasphemous thoughts*, which dart unbidden into the most reverent souls and also *unholy thoughts*, which
torture with a hateful presence the fancy that would fain be pure." Ok. This is Lewis Carroll's introduction talking about all these terrible thoughts,
these naughty thoughts. They're getting his head as a way to stop thinking these terrible things
he goes "against all these some real mental work is a most
helpful ally." In other words; do puzzles because they stop you
At night in bed just before you're going to sleep
(Brady: Alex, Alex, Alex. You've just written a whole book about puzzles.)
I have done, yeah.
- (Brady: What does this say about you?) 
- Well 
- (Brady: Are you having unholy thoughts?)
[laughs]
I'm having incredibly, incredibly, unholy thoughts about all my puzzles all at the same time;
24 hours a day. Yeah
(Brady: So Lewis Carroll invented this puzzle?)
he did invent this puzzle and umm
in the book, he actually writes down the date by each puzzle of the evening, or of the night when he invented each puzzle.
Which is why I told you the day in the first place.
So this puzzle is fantastic, because it's one of those probability puzzles, which means there's a kind of surprising
*Aha* moment but it's incredibly simply stated. What I have: I have a bag and
in this bag, there is a ball. You do not know the colour of this ball
exactly; but we know it's either
red or it's green, and it can be either one.
In fact, we're gonna say that the probability is equal of it being red or green.
So in other words the probability of this ball here is 50%.
Then the puzzle is:
Just say I have a bag with a 50% chance red ball inside it, and I got another red ball.
This ball is red with 100% certainty. We know this ball is red. I'm going to put it in the bag.
I'm then going to
randomly take a ball out.
Randomly taking a ball out, oh, it's red.
What is the probability that the ball that is still in the bag is red?
Ok, now if I were to do this many times I would say: in fact I've got a ball it's red and putting it in,
take one out. It's red. In other words: all I'm doing is putting a red ball in and I'm taking a red ball out.
Putting a Red Ball in, taking a red ball out.
The surprising thing is: that
It changes the probability of the ball that's inside. So what is that probability?
(Brady: So when you just did it once,)
(Brady: what is the probability? Or are you saying what's the probability depending on how many times you do the experiment?)
No, I'm saying what is the probability when I do it once?
I have the ball; I put it in.
To be very clear: I am randomly taking one out
It so happens that the one I randomly selected is red.
What is the chance of this one still being red?
Now most people, if they're not adept at probability problems, will look at this and think, huh?
You just put a red one in and you've taken a red one out. How can it have changed the probability?
Because it seems natural that
the one that's in there that we haven't seen how can its probability have changed? But if you
unpick the way the question is phrased, the probability of this one does change - kind of as if by magic.
You wanna think about it, and then we can maybe go to the solution?
(Brady: Let's do it, have some thinking time)
[Interlude]
(Brady: We're back)
We're back.
Essentially, with these probability puzzles,
we need to look at the sample space and kind of open up and look at the sample space.
Our sample space is look at all the things that are
equally likely to happen. So, we've got a bag. Inside it,
we know that either: there is a red ball or a green ball, with 50% chance one or the other, toss a coin.
So actually the sample space, we know that red ball and
green ball; this is equally likely. Then I'm going to put the red ball in, so
this
is equally likely.
When I take one out, either I'm taking this one out,
or I'm taking this one out, or I'm taking
this one out, or I'm taking
this one out. But we know that the one, I told you I is selecting randomly, these four ones that I take out
are happening at equal likelihood. We can eliminate
the green one being taken out, because I told you when I took a random one out
it was red. In other words: I have either taken out this one, this one, or this one.
If I took this one out,
the one that is still in is red.
If I took this one out, the one that's still in is red.
If I took this one out,
the one that's still in is green. In other words: *still in*. if I take this one out,
still in red. If I take this one out, still in red. If I take this one out, still in green.
We have three different possibilities of a red one going out. On two of them
the one that's still in is red.
So, that's a two out of thirds - 66% chance. So I can redo it with my props. This is red,
this one we don't know what it is. It's either red or green, a 50% chance of being either one.
I put one in, I randomly take one out.
Oh! I randomly take the red one out. The chance of this being red has now jumped up from 50 to 66%
As if by magic, amazing.
Congratulations, Lewis Carroll. You've just invented, by yourself, a fantastic probability puzzle. But more than that, at the same time
contemporaneously, I don't think they knew - umm
I was gonna say plastic workshop
It wasn't. It was Joseph Bertrand, French mathematician, invented a similar setup - essentially the same maths - which is called the Bertrand box
paradox. And then a hundred years later, it became the Monty Hall paradox.
So this is actually, if you do the archaeology of the Monty Hall problem,
this is where it all began, with Lewis Carroll in his Victorian nightcap,
sitting in bed - desperately trying not to think blasphemous thoughts. He comes up with this puzzle
(Brady: It seems to me Alex, the key piece of information)
(here, this moment where you took the green ball out and revealed it to the audience.)
(You've erased that from existence. You've erased that possibility from human existence and the fact you've tampered with the universe in that way - )
Alex: That's, that, that's true. The thing is, what we should have done, is that I should really have
taken one out at random. And sometimes I would get a green one, but we'll just leave those on the cutting room floor. Umm,
so,
often with these probabilities, we need to think about the probabilities that didn't happen.
I'm showing you a possible world
but there are other equally likely possible worlds and the trick - or the clue - is that I say:
I randomly select one and take it out.
Lots of people won't actually register what that means
because they will just see this as only putting one in and taking one out,
but actually we need to realise that; for every time I put one in and take one:
three times I'll come up with a red but one time I'll come up with a green. And we're just - because you didn't see that green
we're just -
forgetting about it, but actually you need to take that green into account as being something that could possibly have happened.
(Brady: And in fact, so when you said "I randomly take a ball out." 
- Yes
(Brady: that's almost kind of like a little lie, isn't it? - Because it's *not* random)
(Because, you're not showing me the times the green came out)
In, in retrospect it's not random - making a video,
but that, that is the slight -
it's not lying, but that's the slight smoke and mirrors of a lot of these probability problems. That's true.
Brady: If you want to find out more about this kind of stuff, then you really should check out Brilliant,
today's episode sponsor. In particular, have a look at these courses on:
Conditional Probability - gonna see some familiar stuff there. And this one here:
Perplexing Probability is also a real classic. Now, All these courses have been
handcrafted by the excellent people at Brilliant. And I like to think of them being a bit like
Nepalese mountain guides, taking you on a journey through learning,
but with all the inside knowledge and knowing all the best routes
and the best way to go, to make the experience the best possible trek.
They've got loads of courses, it's not just mathematics, there's other science too. But a favourite thing of mine are their daily problems.
Here's an example, but you never know what you're gonna get next.
Now you can go right now for free and have a look at Brilliant,
but if you happen to sign up to their premium plan, and get access to all the good stuff, you can get 20% off that
by going to brilliant.org/numberphile
By the way, you don't just have to sign up for yourself -
you can give Brilliant as a gift to someone else. Our thanks to them for supporting this episode.
By the way, if you wanna check out Alex's puzzle book, I'll put a link down in the description.
I'll also put links to his other books. He's done loads.
