"Sleeping Beauty" problem in probability.

**DrunkenArse** · 06-30-2011 09:52 PM

You're abducted for an experiment.

The mad scientists says I'm going to put you to sleep and spin a wheel with 100 numbers on it. If one particular number comes up, I'm going to give you an injection that will make you go back to sleep and erase your short term memory. I'm going to repeat the process of giving you an injection and waking you up 10,000 times and then I'm going to kill you. If any of the other numbers come up, I'm going to let you go.

You wake up. What is the probability that the scientist is going to kill you? To clarify, the wheel only gets spun once.

As far as I know, this is an unsolved problem.

**Replicon** · 07-01-2011 02:17 AM

When you say the wheel is only spun once, you mean once, and then once more after each "injection" right?

**DrunkenArse** · 07-01-2011 02:45 AM

No, I mean once. The outcome decides if you are let go or if you are subjected to the process of being woken up and injected with the memory eraser/sleeping drug 10,000 times.

**sloth** · 07-01-2011 03:23 AM

Oh. I get it.

You wake up.
At this moment, if the unlucky number was spun 10,000 (and one?) spins ago, and the scientist has woken you up ten thousand times already, then THIS TIME he is going to kill you.
What are the chances that the unlucky number was spun (one in one hundred) AND this is the 10,000th time the scientist has woken you up (one in ten thousand)?
I guess you would multiply that? I don't know. I can count to twenty-one if I unzip my pants.

**DrunkenArse** · 07-01-2011 05:56 AM

I really didn't think that I did such a bad job explaining it. I'll try again.

You are informed that you are about to be put to sleep.
After you are put to sleep, a wheel is spun. There are 100 numbers on the wheel. If a given one comes up, say 23, then you will enter the next phase. Otherwise you will be woken up and allowed to leave.

For the next phase, you are woken up and then administered a shot/pill that will put you to sleep and make you forget being woken up. This occurs a total of 10,000 times before you are killed.
There is no wheel being spun at this phase.

So, when you wake up, what are the odds that you are eventually going to be killed as a part of the experiment? (and not later by e.g. a car as you're leaving the scientist's lab or 30 years later in a mugging)

**ty4TheAdventure** · 07-01-2011 06:14 AM

The first thought (and it might be overly simple :O) that came to my mind is that the chances are 1/100. In each result, you would wake up. So the chances of waking up and not dying as part of the experiment outweigh the chances of waking up and being killed.

**DrunkenArse** · 07-01-2011 06:19 AM

That's my thought too. Who wants to post the argument for the other way it could work out?

Remember, that you have just woken up. How many ways can you wake up and live vs. wake up and die? A few physicists actually think that this is the correct answer.

**sloth** · 07-01-2011 04:41 PM

So the scientist wakes up and spins the wheel...
...and then you sleep together?

You're gross, Philosopher.

**tommo** · 07-01-2011 05:15 PM

You've phrased the question horribly man. It just doesn't make sense. Your second explanation is a bit better though.

"This occurs a total of 10,000 times before you are killed."
So he always kills you?
Or are you only killed if the randomly chosen number for each spin never shows up on the respective spin?

So if the number would have to not show up 10000 times....
I think it's a 1/100 chance of being killed.

But wait.... "If a given one comes up, say 23, then you will enter the next phase"
So if the number DOES come up, you enter the next phase?
Then your chances would probably be 99/100 of being killed.

**sloth** · 07-01-2011 08:00 PM

I think I understand.

I am abducted for an experiment.
A mad scientist says that PhilosopherStoned (the lack of exclamation marks in the original post means that the designation "I" in this case refers to PhilosopherStoned) is going to put me to sleep and spin a wheel with 100 numbers on it.
Then the scientists gives me a shot that will make me spin the wheel.
After I spin the wheel then I will number a particular injection 10,000 times and then kill you. If any other scientist is sleeping I will erase my short term numbers.
To clarify, the wheel only gets spun 10,000 times once.

**DrunkenArse** · 07-01-2011 09:05 PM

You people suck.

**tommo** · 07-02-2011 06:03 AM

Originally Posted by sloth

To clarify, the wheel only gets spun 10,000 times once.

LOL That's the part that confused me the most.

PhilosopherStoned - I actually made an attempt to answer it. Was it just way off?

Why don't you just copy the puzzle word for word from wherever you read it?

**Alric** · 07-02-2011 06:37 AM

I think people got confused, because the way you said it makes it absolutely obvious. If you have a 1 in 100 chance of getting the number, then you have a 1 in 100 chance of dying. Being put to sleep a bunch of times after it, has no effect on your chances with the wheel.

**tommo** · 07-02-2011 02:44 PM

Originally Posted by Pan

I might not understand either, but I would think you have a 50/50 chance of surviving after you wake up. Because if you spun the magic number, you're going to die. If you didn't, you're going to live. The amount of numbers that could be spun has no effect on the probability in question, because you already woke up...and once you wake up, there are only two possible fates.

Yeah but the chance of living is significantly lower because one number has to come up instead of 99 other numbers. Or the other way round, coz I don't really get the problem.

**Photolysis** · 07-02-2011 03:58 PM

Edit: Revised

The chances of the victim being killed upon being awoken are quite simple:

The chances of being caught in the cycle (1/100) multiplied by the chances of it being the 10,000th time the victim has been awoken. Given that the victim has no memory of being previously woken, if they are unlucky and subjected to repeated awakenings, the chances of it being the Xth time they've been woken up are all equally likely, so the chances of it being the 10,000th time are 1/10,000

Multiply this together and this gives a probability of 1 in a million of being killed on any individual wakeup,

So, chances you are going to end up dead are 1 in 100.
The chances you will end up dead this time are 1 in 1,000,000

I would think you have a 50/50 chance of surviving after you wake up.[...] there are only two possible fates.

Why do so many people fail at probability and fall into the "all outcomes are equally likely" trap? It's like saying "my chances of winning the lottery are 50-50 because I'll either win or I won't". That's nonsense. One particular outcome is heavily weighted.

**sloth** · 07-02-2011 05:50 PM

Originally Posted by Photolysis

Edit: Revised

The chances of the victim being killed upon being awoken are quite simple:

The chances of being caught in the cycle (1/100) multiplied by the chances of it being the 10,000th time the victim has been awoken. Given that the victim has no memory of being previously woken, if they are unlucky and subjected to repeated awakenings, the chances of it being the Xth time they've been woken up are all equally likely, so the chances of it being the 10,000th time are 1/10,000

Multiply this together and this gives a probability of 1 in a million of being killed on any individual wakeup,

So, chances you are going to end up dead are 1 in 100.
The chances you will end up dead this time are 1 in 1,000,000

Why do so many people fail at probability and fall into the "all outcomes are equally likely" trap? It's like saying "my chances of winning the lottery are 50-50 because I'll either win or I won't". That's nonsense. One particular outcome is heavily weighted.

This is what I think, too. If you look at my very first post on this thread you can see that I worked it out in this way.

**khh** · 07-04-2011 05:33 PM

The reason for the confusion is that there are two ways of interpreting the question and they do not have the same answer.

The first interpretation is "What is the chance of being killed during this experiment?"
In this phrasing we look at the situation as a whole. Basically "Will you be dead or alive at the end?". Since the only thing that defines this is what number the wheel turns up, the chance is 1/100 assuming a uniform distribution.

The second interpretation is "At the time of waking up, what is the chance you now will be killed?"
It is in this interpretation the number of times you will be woken up without being killed comes into play. You can draw up a probability tree (or whatever it's called) to make this clearer.

As is clearly seen the probability of "you die" is only 1/1000000

edit: Or exactly what Photolysis said. Well, at least I included a picture!

**TheAuthor** · 07-04-2011 05:47 PM

The question (in this case) isn't clear on what it's asking by 'will you die'? Is it asking whether you will die from the experiment at all, or is it asking whether you'll die this time that you wake up?

If it's asking whether you'll die from the experiment, then 1/100. Because if you hit that one number, the scientist WILL kill you, it'll just take a long time.
If it's asking whether you'll die this time that you wake up, then 1/10,000,000. Because...well, see Photolysis's post.

**tommo** · 07-05-2011 04:58 AM

Originally Posted by khh

The reason for the confusion is that there are two ways of interpreting the question and they do not have the same answer.

No, I think the confusion lies in the horrible wording for the most part. As well as the fact that most people have no grasp of statistics or probability.

**DrunkenArse** · 07-06-2011 02:21 AM

That's funny tommo. PhantomLegend, Photolysis and khh came in and pretty much just got it. So I didn't word it that badly.

I did mess up a little though and I know this because Photolysis and khh are both working with a different interpretation of the question "are you going to be killed?" than the one I intended. Before you go to sleep the first time, your odds of surviving are 1/00. There's only one way for you to die out of 100 equiprobable outcomes. However, upon waking, your odds of surviving are only 99/10,100 because of the 10,100 possible ways to wake up during the course of the experiment, there are only 99 ways in which to do so and survive.

**tommo** · 07-06-2011 03:27 AM

Well, so did I. I just wasn't sure whether I had the correct answer because of the wording.

10,100 you say?

I don't really get how there are 10,100 ways to wake up, and why your chances are very very slightly smaller upon waking.

**DrunkenArse** · 07-06-2011 04:53 AM

The probability is not slightly smaller but vastly smaller. 99/100 to live before the wheel spins vs 99/10,100 to live after the wheel spins.

There are 10,100 ways to wake up because there are 100 ways to wake up after the wheel spins, 1 for each number. If the bad number comes up, then your short term memory is erased and you are put to sleep 10,000 times. So that's 10,000 additional ways to wake up. Upon awakening, you could be in any of those states.

I personally am highly unsatisfied with this state of affairs.

**TheAuthor** · 07-06-2011 05:07 AM

No. Just...no. When combining probabilities, you multiply, not add.

**DrunkenArse** · 07-06-2011 05:10 AM

uhhhh ..... no?

Because I didn't add at all? Could you explain your objection more precisely?

**TheAuthor** · 07-06-2011 05:15 AM

The 100 and 10,000. From what I can see, you only added the probabilities...

Edit: I probably misread...I have before...but I'm interested in the 99/10,100 thing. Where exactly did you get that?