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**28 members have voted**

Quote:CalderThe math in these questions is always beyond me, but don't you mean Chuck Berry?

Nope. All due respect to Chuck Berry, but I identify with celebrities who hit their prime in the 70's or 80's. Chuck Barris had a huge mark on the 70's culture. I'd hate to calculate how much time I spent watching shows he starred or produced.

The coin is only tossed 1 time, right after she is put to sleep. There's a 50% chance that a heads was tossed and she is woken up on Monday. And, a 50% chance a tails was tossed and she is awoken on Monday and Tuesday.

Sleeping beauty will not remember being awoken on Monday or Tuesday. She will think she slept straight through until Wednsday.

The coin is only flipped 1 time, so the results of the flips should be 500,000/500,000 in a million flips. Unless using a simulator, where it may be 550,000/450,000. But, ignoring that, the results would be 50/50 for the coin flips.

For the 1/3 people. The coin is only flipped 1 time, on Sunday night after sleeping beauty goes to sleep. In this case, the chance of being awoken one time is 50%, and the chance of being awoken 2 times IS ALSO 50%.

So, if there's 1.5 million awakenings, 33% being a heads, and 66% after a tails. Then the coin flip was 50/50. 50% heads and 50% tails.

So, the correct answer for sleeping beauty is 50%. Even if she woke up more often on tails, the coin flip results were 50/50.

Quote:Wizard

Let's say Sleeping Beauty is on a game show. They start by giving her $1,000,000. Then she agrees to go through this experiment 1,000 times. She is required to guess the same probability every awakening. A tally will be kept of the heads and tails leading to each awakening. At the end of the 1,000th experiment there should be close to 1,500 data points, because there are an average of 1.5 awakenings per experiment. The ratio of the "heads" data points to all of them will be calculated. Then it will be compared to Sleep Beauty's prediction. For every percentage point Sleep Beauty is off, $10,000 will be deducted from her million dollars. What probability should be render at every awakening?

In this case, Sleeping Beauty, should answer 33% every time she is awoken. Because she will have awoken twice as often when a tails lands.

Quote:WizardShe can answer whatever she wants at each awakening. However, she would remember nothing, so allowing changing of strategy doesn't help that I can see.

It is a paradox because there is no accepted answer.

Since what is being asked is a bit vague, let me offer a rewording, which the 1/2 camp may take issue with.

Let's say Sleeping Beauty is on a game show. They start by giving her $1,000,000. Then she agrees to go through this experiment 1,000 times. She is required to guess the same probability every awakening. A tally will be kept of the heads and tails leading to each awakening. At the end of the 1,000th experiment there should be close to 1,500 data points, because there are an average of 1.5 awakenings per experiment. The ratio of the "heads" data points to all of them will be calculated. Then it will be compared to Sleep Beauty's prediction. For every percentage point Sleep Beauty is off, $10,000 will be deducted from her million dollars. What probability should be render at every awakening?

Per the bold.

I don't know the definition of a paradox n stuff. But I thought a paradox had 2 (or more) answers where when you give one answer, that can't be correct because of some X, so the only appropriate answer is answer B. But answer B isn't correct because of Y, so only answer A is appropriate....but A isn't appropriate because of X......round and round it goes.

Or in other words -- there is more than 1 "correct" answer yet there is no correct answer. Or something like that.

Like the Indian who shoots an arrow at a tree at a steady rate of 10 ft/s. If he's 20 feet away, after 10 feet it'll take 1 second to travel. Traveling half the previous distance, it'll take 0.5 seconds to travel 5 more feet. (Now 15 feet @ 1.5 seconds) Then 0.25 seconds to travel 2.5 feet. (Now 17.5 feet @ 1.75 seconds) Then 0.125 seconds to travel 1.25 feet. Etc. The numbers never add up to a full 20 feet, even though they may get infinitely close to 20, won't reach 20, yet the arrow clearly hits the tree after 2 seconds.

This is a different puzzle as all Sleeping Beauty has to do is estimate the ratio of Heads used to wake her up.Quote:Wizard...A tally will be kept of the heads and tails leading to each awakening...The ratio of the "heads" data points to all of them will be calculated. Then it will be compared to Sleep Beauty's prediction...

Since she can't change her answer she can give this at the beginning of the trial. She knows that, on average, there will be a similar number of heads (each Head creates one data point "H") and tails (each Tail creates two data points "T"). Thus on average there will be 1/3 "H" and 2/3 "T".

Another way of looking at it.

If every time they wake her up they either put an "H" ball in the bucket or a "T" ball in the bucket. Each game will either (Heads) result in one "H" ball or (Tails) two "T" balls in the bucket. For 1000 games there should be, on average, 500 "H" and 1000 "T", so the ratio is 1/3. This is what she should guess.

ps - I can't see the paradox with this.

Quote:JyBrd0403Then the coin flip was 50/50. 50% heads and 50% tails.

The question isn't asking what the coin flip results were.

Quote:charliepatrickThis is a different puzzle as all Sleeping Beauty has to do is estimate the ratio of Heads used to wake her up.

Since she can't change her answer she can give this at the beginning of the trial. She knows that, on average, there will be a similar number of heads (each Head creates one data point "H") and tails (each Tail creates two data points "T"). Thus on average there will be 1/3 "H" and 2/3 "T".

The 1/2 camp evidently doesn't understand your argument, which I agree with. I was trying to put it another way, which may have only confused the issue more.

Quote:

Another way of looking at it.

If every time they wake her up they either put an "H" ball in the bucket or a "T" ball in the bucket. Each game will either (Heads) result in one "H" ball or (Tails) two "T" balls in the bucket. For 1000 games there should be, on average, 500 "H" and 1000 "T", so the ratio is 1/3. This is what she should guess.

You're preaching to choir with me.

Quote:ps - I can't see the paradox with this.

In the future, if I write about this, I'll call it a puzzle instead. Other sources refer to it as a "paradox" because it seems counter-intuitive that the answer is anything about 1/2. However, I don't see how the 1/2 side holds any water. Much like Alan's two-dice puzzle, just because so many people are wrong, doesn't mean the right answer should have any doubt.

I would consider that a substantial "rewording." Its as if you claim you are going to flip a fair coin numerous times but will count each "heads" double what you will count a "tails" when you tally the results. Then you act as if that changes what a "perfect logician" should predict as the probability of whether a toss is heads or tails.Quote:WizardSince what is being asked is a bit vague, let me offer a rewording, which the 1/2 camp may take issue with.

Did I miss something somewhere?

How's this for a rewording:

I will flip a fair coin numerous times. If a result is tails, I will record "1" in the tails column. If the result is heads, I will record a variable number in the heads column. That number will be biased, but you will have no information as to how I determine the bias. Your assignment is to determine which column will receive the higher total at the end of the flips. How do you come up with the answer if you are a "perfect logician"?

Seems like a wasted effort, not what I consider a paradox.

Then again, I'm always wrong on these apparently simple odds questions!

EDIT: I didn't read the rewording or anyone else's answers. My answer relates to the OP.

The girl (perfect logician) will be informed fully of the rules. She will then be placed in a trace, and a fair coin will be flipped one time. If it is heads, she will be executed in her sleep. If it is tails, she will be awakened and asked what the probability was for a heads result.

Under that wording, I think her answer would/should be: "The probability of heads was and is 50%. In this particular trial, the actual result appears to have been tails."