## View Poll Results: What should the woman answer?

Voters
12. You may not vote on this poll
• 1 in 2

7 58.33%
• 1 in 3

3 25.00%
• Don't know! It's a paradox!

2 16.67%

1. ## Probability Paradox of "Sleeping Beauty"

I read this interesting puzzle the other day... and thought I'd share it with you to boggle your minds...! I'd love to hear your thoughts!

For unknown reasons, a woman is told that she needs to take a drug which will wipe her memory and put her to sleep.

The doctor says that he will toss a coin. If it lands heads, he will administer the drug and a few minutes later she will wake up with no memory of the past. If it lands tails, he will do exactly the same, but then administer the drug one more time so she falls asleep for a second time, again waking with no memory of the past.

The woman wakes up and the doctor asks, "What are the chances that the coin landed heads?"

2. I believe 1 in 3 is the correct answer, though if I recall correctly it is still a point of academic debate.

I can't explain why particularly easily. The difference is due to conditional probability; everybody agrees that the chance of a fair coin landing heads is 1/2, But the question is not "What is the probability of throwing heads?" the question is "What is the probability of heads being thrown given she has been woken and asked?" This conditional can change the probability if you accept that the scenario provides new information.
People will be familiar with conditional probability in the Monty Hall problem: your confidence that the door you picked hides a prize changes when you are presented with new information, in this instance one of the non-prize doors being revealed.

What new information the scenario provides is questionable but the key is always that the woman is unable to tell whether she is being asked for the first time or for the second time. I wrote down all the conditional probabilities e.g. P(Heads | 1st ask), P(1st ask | Tails) etc. then after working through the algebra I obtained P(Heads) = 1/2 P(Tails). As P(Heads) + P(Tails) = 1 then P(Heads) = 1/3.

3. Originally Posted by tiny
The woman wakes up and the doctor asks, "What are the chances that the coin landed heads?"
"what coin? who're you? what the fcuk am i doing here? wtf? where are my clothes? WTF?!!!!!!"

4. Well, only 1 time only tossing the coin, and it got 2 sides, so 1 in 2 obviously.

Would have been kinda interesting if he would flip the coin every time again after it lands on tails, or every time when she awakes again ^_-
That's what I thought at first while reading.

5. Originally Posted by WoodlandWanderer
I believe 1 in 3 is the correct answer, though if I recall correctly it is still a point of academic debate.

I can't explain why particularly easily. The difference is due to conditional probability; everybody agrees that the chance of a fair coin landing heads is 1/2, But the question is not "What is the probability of throwing heads?" the question is "What is the probability of heads being thrown given she has been woken and asked?" This conditional can change the probability if you accept that the scenario provides new information.
The way I see it (after a lot more thought), there was a 1 in 2 chance of the coin landing tails. And on that branch of the tree, two equally possible points where the woman could have woken up.

The fact that there are three possible awakenings doesn't mean that there's a 1 in 3 chance because each awakening is not equally likely. If the woman knew that the coin had landed tails, there are still two possibilities as to whether she has just woken up for the first time or the second. She has a 1/2 chance of being right. Multiply that by the 1/2 chance of the coin landing tails, and each awakening on the "tails" branch is 25% likely to be the present one, while the awakening on the "heads" branch is still 50% because, it was decided by a simple coin toss.

"what coin? who're you? what the fcuk am i doing here? wtf? where are my clothes? WTF?!!!!!!"
Ha ha ha! Oh yeah... I didn't think of that! D'oh!

Maybe I should have said that the doctor explains after she wakes about the amnesiac drugs and the coin toss process without revealing what the outcome was. Or something...

6. I teach probability to 7th and 8th graders, and I know that each time you toss the coin, you are multiplying 1/2, so two tosses have a 1 in 4, or 25% chance, not 1 in 3. Clearly I'm missing something.

7. No chance. If she remembers the discussion at all, he obviously never dosed her, meaning he obviously didn't flip the coin.

Alternatively, the odds would be 1/1 it was head, as she would've forgotten the initial discussion if dosed twice, correct?

Statistically speaking - There's no absolute answer, as there are too many variables to interpret. More information would be required to give a definitive result.

8. Originally Posted by tiny
I read this interesting puzzle the other day... and thought I'd share it with you to boggle your minds...! I'd love to hear your thoughts!

For unknown reasons, a woman is told that she needs to take a drug which will wipe her memory and put her to sleep.

The doctor says that he will toss a coin. If it lands heads, he will administer the drug and a few minutes later she will wake up with no memory of the past. If it lands tails, he will do exactly the same, but then administer the drug one more time so she falls asleep for a second time, again waking with no memory of the past.

The woman wakes up and the doctor asks, "What are the chances that the coin landed heads?"

Sorry, I need a bit of clarity, as that's important in questions like this. To confirm, is this correct?

• There is only one coin flip (i.e. it's not "flip a coin, and if it's tails flip again").
• The doctor asks the question at the very end of the process (i.e. once the doctor asks the question he's not going to drug her again).

So to be precise about it, there are two possibilities:

1. The coin flip is heads. The doctor administers the drug, she wakes up, and then he asks the question.
2. The coin flip is tails. The doctor administers the drug, she wakes up, the doctor drugs her again, and then he asks the question.

If you draw a probability tree, there's just one event. You don't even need Bayes Rule - it's 1/2.

Next up - Monty Haul problem, anyone?

9. Originally Posted by dogboy
I teach probability to 7th and 8th graders, and I know that each time you toss the coin, you are multiplying 1/2, so two tosses have a 1 in 4, or 25% chance, not 1 in 3. Clearly I'm missing something.
Ah... but the coin is tossed only once. Then she is drugged once or twice (according to whether it's heads or tails).

Originally Posted by Eulogy
No chance. If she remembers the discussion at all, he obviously never dosed her, meaning he obviously didn't flip the coin.
D'oh! It doesn't happen like that! It's just a theoretical puzzle, not a "trick". I didn't accurately remember the problem and had reformulated it in my mind. Just imagine that the doctor explains what has happened to her up to the point just before he tossed the coin. So she knows about the experiment, but not whether the coin toss resulted in heads or tails.

Originally Posted by LittleAcorn
Sorry, I need a bit of clarity, as that's important in questions like this. To confirm, is this correct?

• There is only one coin flip (i.e. it's not "flip a coin, and if it's tails flip again").
That's right. The doctor tosses the coin once to decide whether to drug the woman once or twice.

Originally Posted by LittleAcorn

• The doctor asks the question at the very end of the process (i.e. once the doctor asks the question he's not going to drug her again).
Ah, no -- all the woman knows is that she has just woken up. She could be in any of the three "awakening situations". (Although as ade said, she'd have to be reminded of the nature of the treatment/experiment if the drug induced total amnesia.)

There are three possibilities:
1. The coin flip is heads. The doctor administers the drug, she wakes up, and then he asks the question.
2. The coin flip is tails. The doctor administers the drug, she wakes up, and then he asks the question.
3. The coin flip is tails. The doctor administers the drug, she wakes up, the doctor drugs her again, and then he asks the question.

Originally Posted by Eulogy
[T]here are too many variables to interpret. More information would be required to give a definitive result.
Okay... I've just found information about the original problem that was described, and it's a little bit different to mine (and I don't understand the mathematical notation), but here you go:

Sleeping Beauty problem - Wikipedia, the free encyclopedia
Lecture 23 &mdash; Notre Dame OpenCourseWare

10. What's happening here is that an independent outcome has a variance, one that id dependent. I think in reality, they are taking the average between two of the four outcomes, thus making three.

I keep wondering, if while she's under, the doctor takes sexual advantage of her. What are the chances she finds out and shoots him in the head? Remember, she could shoot him in the crotch, or other fair body parts. Don't you love it when we call it a "fair coin"...haha. That's because some coins and dice are weighted, thus determining an outcome. By waking Sleeping Beauty a second time, the good doctor plays with the independent outcomes, making one of them dependent upon the original, two outcomes, in my humble confused opinion.