Math Paradox

[MathGuy]

Here's a pretty interesting math paradox:

Sleeping Beauty is going to take part in an experiment. Without showing her on Sunday night, the scientist is going to flip a fair coin. Then, he will put her to sleep with a drug that erases her short-term memory, so that when she wakes up she doesn't remember if or how many times she's been awakened before.

On Monday morning, he'll wake her up. If the coin turned up heads, he'll only put her to sleep that once, but if it came up tails, he'll put her to sleep again Monday night and wake her up one more time on Tuesday morning. Each time he wakes her up, he asks her the question, "How likely do you think it is that today's Monday?" What should she answer?

EDIT: The question should be, "How likely do you think it is that the coin came up heads?"

(Note: there are two commonly given answers to this problem; although most people prefer one of the answers. If you think the answer's obvious, you may want to think about it a little more, because there are many smart people who disagree on the answer. )

 

[Claudya]

Does she know her memory has been tampered with? (I'd guess so because otherwise the paradox wouldn't make much sense at all, so I'm gonna assume she fully understands the experiment?)

So I myself just woke up, I have a day off so I slept in and didn't drink enough coffee yet. The part of my brain that is responsible for higher thinking isn't awake yet. Chances are good this is a thursday today, but I can't give you numbers.
On topic...
my very first guess would have been she has a 50% chance to wake up on monday and 50% to wake up on tuesday because a fair coin would give her a 50% chance of being put back to sleep.
But that feels kinda wrong. Monday would be a much more likely answer. She will definitely have a wake-up on Monday, but has only a 50% chance to have a wake-up on tuesday.
If that experiment would be repeated many times, say 100 times, she would have 100 monday wake-ups, but would only awaken on half of the tuesdays.
So her chances for the day beind a monday are .66 or 2/3 (and consequently .33 or 1/3) for tuesday.

But that answer is still suspicious because there ain't much of a paradox to it, so maybe I have missed something....
I need coffee really bad.

 

[Contrarian]

The odds of it being Monday are 1 in 7. Regardless of whether you remember which day it is.

 

[Claudya]

The odds of it being Monday are 1 in 7. Regardless of whether you remember which day it is.

I will remember your words and take comfort from them if I ever wake up too hung over to remember what day it is. Maths is the natural enemy of confusion. :D

But as I understood the paradox, the experiment will go on for two days at most? So when the coin lands heads up, she will be sent home on Monday, when it lands tails up she'd get her amnesia drug, be put back to sleep and awakened on tuesday... and then sent home?

Or did I get that wrong? :help

 

[Sparkey]

I did a Google search on "Sleeping Beauty Paradox" and found:
Quoted from:Measure of Doubt

“I’m going to put you to sleep tonight, and wake you up on Monday. Then, out of your sight, I’m going to flip a fair coin. If it lands Heads, I will send you home. If it lands Tails, I’ll put you back to sleep and wake you up again on Tuesday, and then send you home. But I will also, if the coin lands Tails, administer a drug to you while you’re sleeping that will erase your memory of waking up on Monday.â€
Quoted from:Measure of Doubt

The "paradox" comes when the coin is tossed out of Beauty's sight on Monday after being woken.

So when she wakes up, she doesn’t know what day it is, but she does know that the possibilities are:

• It’s Monday, and the coin will land either Heads or Tails.
• It’s Tuesday, and the coin landed Tails.

We can rewrite the possibilities as:

• Heads, Monday
• Tails, Monday
• Tails, Tuesday

I’d argue that since it’s a fair coin, you should place 1/2 probability on the coin being Heads and 1/2 on the coin being Tails. So the probability on (Heads, Monday) should be 1/2. I’d also argue that since Tails means she wakes up once on Monday and once on Tuesday, and since those two wakings are indistinguishable from each other, you should split the remaining 1/2 probability evenly between (Tails, Monday) and (Tails, Tuesday). So you end up with:

• Heads, Monday (P = 1/2)
• Tails, Monday (P = 1/4)
• Tails, Tuesday (P = 1/4)

So, is that the answer? It seems indisputable, right? Not so fast. There’s something troubling about this result. To see what it is, imagine that Beauty is told, upon waking, that it’s Monday. Given that information, what probability should she assign to the coin landing Heads? Well, if you look at the probabilities we’ve assigned to the three scenarios, you’ll see that conditional on it being Monday, Heads is twice as likely as Tails. And why is that so troubling? Because the coin hasn’t been flipped yet. How can Beauty claim that a fair coin is twice as likely to come up Heads as Tails?

 

[MathGuy]

Sorry, I sort of messed this up! The questions she's supposed to be asked is, "How likely do you think it is that the coin came up heads?" (If the coin came up heads, she's only awakened once.) Also, yes, is aware of the setup of the experiment, so she knows that it can only be Monday or Tuesday. The paradox is that there are two commonly accepted answers, and there hasn't been a general consensus on what the answer should be.

 

[Gordon]

I think you did the math wrong with the 1/2 1/4 1/4. It is combining the chance of 50% with 100% for tails. Not one event but two. Same with heads combining 50% with 0%. Then you'd have to figure the event of a second day is only 50%. I have no idea how to do this.

 

[Contrarian]

Yes,

But what day is it when Prince Charming slays the evil scientist, and kisses sleeping beauty.