Today we are talking about Sleeping Beauty in
the form of the Sleeping Beauty Paradox. It goes as follows: Sleeping Beauty is going
to be put to sleep on a Sunday - I should clarify Sleeping Beauty has agreed to do this
and knows all of the details of the experiment, this is important. Sleeping Beauty is put to sleep
on Sunday. A coin is then flipped; if the coin is tails we are going to wake her up on Monday, we
then ask her some questions, and then she's put back to sleep with some amnesia inducing drug so
she has no memory that she was ever awake. This is again very important to the setup. And then
we also - same thing - wake her up on Tuesday, ask the same questions, put back to sleep. And
again she has no idea. If the coin was heads, she's woken up on Monday only, asked questions,
perhaps with the same amnesia inducing drugs. Sleeping Beauty is put to sleep, we flip
a coin. If it's tails we wake her up on Monday, ask her some questions, put her back
to sleep. She doesn't know it was Monday, she doesn't know what happened to the coin, she
has no idea when she wakes up what's going on. Put back to sleep with no memory of
that incident. She's woken up on Tuesday, same thing, ask questions, put back to sleep;
has no idea it's Tuesday or what's happened. Or if it was heads we just wake her up the once
and then ask her the questions, put her back to sleep, nothing happens on the Tuesday; and in both
cases the experiment ends on Wednesday. So she's woken up and she's free. And she knows all of this
before the experiment begins, so she is aware that she may be woken up twice or once depending on
the outcome of the flip of that coin.
- (Brady: Tell you) (what, as a recent father, the idea of sleeping from
Sunday till Wednesday sounds lovely)
- Okay Sleeping Brady, not Sleeping Beauty.
- (Anyway, okay, okay. On Wednesday she's woken up)
- Experiment ends. Now, when I say here ask questions, there's a key
question that we're going to ask Sleeping Beauty. And the question is we're going to ask
her, what does she believe is the likelihood or the chance or the probability that the coin
was a head? On the situation she is in at that moment of having been woken up. So Sleeping Beauty
has woken up on a Monday or a Tuesday, she has no idea, she has nothing that can help her figure
out which day it is, there's no secret clock or calendar, she has no idea. She's just woken up and
asked, what do you think the probability was that we got a head on that coin? And of course the idea
being she could have been woken up on a Monday and it could be the first of the tails wake ups. It
could be the Tuesday which means it would have had to be tails or it could be the Monday but it
could have been the only wake up on the Monday. But because her memory is wiped after each wake up
she doesn't know if it's once or twice or anything. So that's the question, what's the probability
that the coin was a head? Now there are two both well-supported trains of thought as to
what the answer to this question is - and I'm sure everyone watching has their own opinion
right now of what they think the probability is. (Well I'll try and figure it out. She's asked the
question either- is she asked the question when she) (wakes up as well?)
- No not at the end. No no no no that's just-
- (Okay so she's asked the) (question either two or three times? No, she's
asked the question either one or two times?)
- Yes (So) (I reckon...oh let me think for a
second. Two thirds of the time) (it was a- I was going to say two-thirds
of the time it was a tail. But she's) (asked the question. So she's asked
the question twice on Monday) (and she's asked the question once on a Tuesday; but
she's asked the question less often on a Tuesday.) (So- I don't know. What was the
probability that the coin was a head?) (Oh,50/50?)
- Well valid answer right? Option one,
it's called the Halfer approach to the problem, is literally I flipped a coin on Sunday
who cares when Sleeping Beauty was woken up? I flipped a fair coin, it's a 50/50 chance,
the probability of it being a head is a half. Because if I flip a coin right now it's a half and
the coin was flipped before she was woken up. She doesn't learn any new information by being woken
up so she just says, okay I've learned nothing new, I can't apply my Bayes theorem conditional
probability of updating your hypothesis with new information; she hasn't learned anything new.
- (Monty Hall hasn't told her anything she didn't know) - Exactly! But Sleeping Beauty has no new information because of
this sort of waking up and not knowing anything and the amnesia effects of the drugs. So she has
no new information; therefore it's the same as the probability always was: it's a half. That is the
argument for the Halfers in this problem. There is another potential answer and that is that it's
a probability of one-third - the Thirder approach. So Sleeping Beauty does not know what day she has
woken up. Suppose we are Sleeping Beauty and we can say, okay well let's assume it was a tails, what
can I learn in that particular situation? Now if we got a tails what does that tell us? That tells
us we are going to be woken up twice. Therefore when we are woken up and asked this question,
having assumed that it was the tails in our own head, we are like okay this could be either
Monday or Tuesday and they are equally likely, there's nothing to distinguish between those cases.
So we can say the probability of it being a Monday, given it was tails because we assumed it was tails,
is therefore equal to the probability of it being a Tuesday - again given I assumed it was a tail.
It's equally likely it could be in one of these two situations. Now we have law of conditional
probability tells us the probability of A given B is the probability of A intersect B divided by
the probability of B. So we can apply this here and say, so this one is the probability of Monday and
tails divided by the probability of being tails; and this one is the probability of Tuesday and
tails divided by the probability of being tails. These two are equal so it doesn't matter what's on
the bottom, multiply through. Having assumed it was tails, our conclusion we can draw is that these two
things are equal because they're all on the same side of the equation. So we can say the probability
of Monday and tails is equal to the probability of Tuesday and tails. Okay, now that was assuming
that it was tails. What if we instead assume it is Monday. Sleeping Beauty wakes up, has no idea what's
going on, so she's like all right well let's just assume it's a Monday - what can I learn about that
situation given I'm assuming it's a Monday? Same kind of idea: we could be either this one, the first
wake up for tails, or the only wake up for heads. So we've assumed it's a Monday so that means the
probability of it being a tails given it's Monday, this case, indistinguishable from this case.
The probability of it being a heads given we know it's Monday - or we've assumed that it's
Monday. Same thing, conditional probability law, tails of Monday, well that's the probability of
tails and Monday divided by probability of being Monday. This one is the probability of heads and
Monday divided by probability of Monday. These two are equal, they both have the same denominator
so we ignore it, so therefore the probability of tails and Monday equals the probability of heads
and Monday. Now we have this equation and we have this equation. So we've got Monday and tails; tails
and Monday, they're the same thing. In both cases it's a Monday and tails is flipped. So this one's
equal to Tuesday and tails and same thing is also equal to heads on Monday. So what we've done here
is actually shown that these are like equal, all three things are the same, and all three of them
are exactly case 1, case 2, and case 3 of our original list. We made some assumptions, did
some conditional probability, and it allowed us to arrive at the conclusion that all three of
our situations actually are equally likely to occur. The total probability has to be 1.
These are the three possible events, they are all equally likely, their total probability
has to be 1 - it's the law of total probability. The only solution then is that each individual
event has a probability of one-third, because they're all equally
likely and there's three of them. So we say okay, so the probability
is a third for each situation. So it's it's a probability of a third that it's a
Monday and tails was flipped. The probability is a third that's it's a Tuesday with tails flipped,
and probably a third that it was a Monday with heads flipped. So what's the probability
that the coin was heads? One in three. One of the three situations has a head; they're all equally
likely, so the probability of the coin was a head is a third - and that is the Thirder argument.
- (No but there's a change. I think there's a slight change) (there just the last minute. Saying those three
things are equally likely is not say- it's not) (the same as saying: is it likely the coin was a
tail or a head? Because once the the coin is flipped) (a whole- whole different paths are
opened up, and different things are) (done. And it's those paths and things you're
doing the probability of, not the coin itself.) (So we're we're not- we're asking Sleeping Beauty
not was it a head or a tail but which sort of) (mashup combination of things that happened
is playing out at the moment?)
- This is- okay I- Both of these are valid arguments. Both of
these have supporting mathematical statements using Bayesian statistics, using conditional
probability; you can argue for both of them in different ways. Both of them
are accepted by mathematicians, that is why it's a paradox. Because it feels like-
you you can argue successfully for either of them and you can argue against either of them. Now
there is an interesting extension which might help to convince you at least Brady, about the one-third
situation. And that's called the Rip Van Winkle extension to the problem. If we get a head Sleeping
Beauty is woken up once - same rules, same experiment, but just once. If we flip and get a tail Sleeping
Beauty will be woken up 999 consecutive times. It is the same setup as we had but instead of being
once for heads and twice for tails it's now 999 times, big old sleep; hence the name Rip Van Winkle.
Now if you follow through the Thirder argument; or even if you just think about it if you're in
the position of Sleeping Beauty: if I wake up as Sleeping Beauty there are 1,000 potential events
or 1,000 potential situations I could be in. And in only one of those possible situations did a head
come up on that coin. So if I'm a betting princess I am definitely saying that I've got to go with tails.
- (Here's- here's the thing though then, that's) (what I mean about my paths. When you wake her
up you're not asking her: was it a head or a) (tail? You're asking her: are you on the path that
resulted from a head or are you on the path that) (resulted from a tail? Because yes of course it's
more likely she's on the path that resulted from) (a tail because she gets woken up multiple times.
Like if I said to her, every time you're asked the) (question, if you answer it correctly I'm going to
put a thousand dollars in your bank account)
- Yeah, you're gonna say tails.
- (You will say tails because you're woken up twice with tails and you're woken up once for heads.) (So I agree if the question is which pathway
are you on? Yes I'm on the tails pathway. But if you) (ask, when that coin was tossed on Sunday while
you were asleep, was it a head or a tail? Well-) But does that not then come back to the question
of- so you could say, which path are you on? And as you're saying, it's like well clearly I am on
the tails path. So then could I not then say, okay so you're saying it was much more likely that I
got a tails on that coin? I.E the probability that coin was a tails-
- (No. It's more likely I'm on the
path that resulted from the tail, yeah yeah) And I think- I'm really glad you picked up on this
because that is kind of the the philosophy extension to this problem. So I- researching this,
like I first heard about this from one of my students in fact and I just thought it was such a brilliant
niche little problem that just really gets you thinking. And again both solutions are actually
equally credible from a mathematics standpoint, but I think it does come down to the sort of like
interpretation of what is it you are actually asking Sleeping Beauty. The original problem was
written up in April 2000 by Elga, E-L-G-A. And I think there the original statement of the problem was,
you ask Sleeping Beauty what is your credence that the coin was a head or like- almost like
what's your strength in belief? So it kind of- you know, maybe by me using the word probability
is- but that's kind of how it's interpreted in a mathematical way. But I think the resolution of
the paradox does come from that way of thinking about what are we really asking Sleeping
Beauty rather than what is the probability that the coin was a heads or a tails.
- (Yeah. Sleeping Beauty would be correct more often if she said tails.) Exactly. But-
- (But she's given more chances to
be correct if it's a tails as well. Like the whole) (game is rigged that way. Yeah yeah okay.)
- So it feels as though here you would have probability of being a head, you would say well only one in a
thousand events are going to be heads so I have like very very small belief in heads. So I'm saying
I think there was a one in one thousand chance that the coin was a head even though it was a fair
coin, which is just quite hard to- paradox This episode was brought to you by Jane Street, people
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more about their internships, jobs, everything else. Super exciting, we think they're going
to be really common - they're not. The next one is at 16,470. So I'm definitely not writing out 16,000 digits of pi, that that's that's a job for
the viewer. And then after that we then jump up to 44,899. You've accelerated, your speed has
changed with respect to time or you decrease your speed, you've decelerated. So that's what
the first term is going to describe and then we also need the mass. You think of
mass in this situation to be a density
