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[goblolo]

1. The coin came up heads, and it's Monday.
2. The coin came up tails, and it's Monday.
3. The coin came up tails, and it's Tuesday.

yes, but the probabilities are the following:
1. The coin came up heads, and it's Monday. 50%
2. The coin came up tails, and it's Monday. 25%
3. The coin came up tails, and it's Tuesday. 25%

it is because when you flip the coin, you have 50% for heads and 50% for tails.
If you have heads, when you are awakened, it is 100% monday, so we have 50% * 100% = 50% that it is monday and heads.
If you have tails, when you are awakened there could be monday and tuesday with 50% relative probability. Thus you get 50% * 50% = 25% for each of 2. and 3. occasion.
Taking into account that 1. is heads and 2. + 3. is tails, we have 1:1 for heads/tails

[Soadreqm]

2/3rds. If she's awake, one of three things happened:

1. The coin came up heads, and it's Monday.
2. The coin came up tails, and it's Monday.
3. The coin came up tails, and it's Tuesday.

Hence, 2:1 odds.

I'm not quite convinced yet. Those three things aren't equally likely to happen. Suppose the following scenario: Instead of being interviewed twice with a mind-wipe inbetween, she's placed into an eternal Groundhog Day loop of mondays forever, her memories erased every night. (If the coin comes up heads, she's woken up and interviewed once, without time shenanigans, as before.) Now, there are infinite cases where the coin came up tails, and just one where it came up heads, so the probability of it having come up tails when Sleeping Beauty is woken is 1. The probability would, by this logic, remain the same if instead of a coin toss, we rolled a d100 and put her to a time loop on a one. Or a d10^100. As long as there is a nonzero chance that she's been put into a time loop, she can say with perfect certainty that she is in a time loop.

Applying this logic outside the thought experiment, we can conclude that the most likely explanation for it being Friday today in the real world is that the world is stuck in an infinite time loop of Fridays. >:]

Or to invert the example: An infinity-sided die is rolled. On any number but one, Sleeping Beauty is put to sleep and awakened on the next day, and the experiment ends. On one, Sleeping Beauty is put to sleep and awakened on the next day, mind-wiped, put to sleep again and reawakened on the next day. Since there are three cases
1) The die came up one, and it is day 1
2) The die came up one, and it is day 2
3) The die came up something other than one and it is day 1
The probability for the literal infinity-to-one chance having happened is 2/3.

Also I spent too long typing this and goblolo said the same thing.
Whatevs.

[alway]

1. The coin came up heads, and it's Monday.
2. The coin came up tails, and it's Monday.
3. The coin came up tails, and it's Tuesday.

yes, but the probabilities are the following:
1. The coin came up heads, and it's Monday. 50%
2. The coin came up tails, and it's Monday. 25%
3. The coin came up tails, and it's Tuesday. 25%

it is because when you flip the coin, you have 50% for heads and 50% for tails.
If you have heads, when you are awakened, it is 100% monday, so we have 50% * 100% = 50% that it is monday and heads.
If you have tails, when you are awakened there could be monday and tuesday with 50% relative probability. Thus you get 50% * 50% = 25% for each of 2. and 3. occasion.
Taking into account that 1. is heads and 2. + 3. is tails, we have 1:1 for heads/tails

Yay Bayesian Reasoning! The basis of modern Artificial Intelligence and modern scientific reasoning in general.

[IronyOwl]

Yeah, you're right, it's even odds.

Also I spent too long typing this and goblolo said the same thing.
Whatevs.

Yours was a much more instructive example, though. goblolo approached it from the wrong direction, ie what are the odds we'll flip a coin and get this result, not what are the odds we've gotten this result now that we know the outcome. The answer ended up being the same because the outcome didn't preclude or favor any results, but it's still the wrong end of the problem.

[Ogdibus]

.

[Criptfeind]

Hum, I was not sure about this. So I went to random.org and had them flip 200 coins. Imagining each as a separate experiment. I got 104 heads.

So she is woken up 104 times and the correct answer is heads. And she is woken up 192 times and the correct answer is tails.

Conclusions: Undrawn as it is is 7:30 in the morning with no sleep. But it was fun to count.

Edit: Oh, I think I read it wrong, the chance that it came up tails, not the chance that it was tails. Or something. Maybe.

[goblolo]

goblolo approached it from the wrong direction, ie what are the odds we'll flip a coin and get this result, not what are the odds we've gotten this result now that we know the outcome.

After thinking a bit I got to the point, that there is 2:1 chance, because it is the situation from the sleeping beauty's point of view. My previous post showed how to aquire the chance of head or tail as a result of tossing the coin, not the coin position after awakening. Yes, the chance for heads/tails is 1:1, that's simple, but the amount of awakenings with tails up will be twice a number of "head up" awakenings.


Strange thing! Lets put a bird counter to the feeder. If we got pigeon, he will eat 2 seeds, sparrow will take only one. There are equal amount of sparrows and pigeons. Feeder can hold only 1 bird at the time.
We have 1 seed there. For this seed there is 50% chance to be eaten bu pigeon and 50% chance to be eaten by sparrow.
If we have 2 seeds, then the situation changes. The random seed can be eaten at first (50%) or at last (50%). In the case of first one to be eaten we can get 2 occasions: pigeon has come (relative 50% and absolute 25% chance) and 25% for sparrow. If the seed happens to be last one, we have absolute 25% that pigeon ate first seed, and he will eat the secon one, and absolute 25% that sparrow ate the first seed, thet there will be 12,5% for pigeon and 12.5% for sparrow to eat the second seed. After counting all the occasions we'd get 62.5% for pigeon and 37.5% for sparrow to eat chosen seed!

[Scoops Novel]

You XKCD reading bastard goblolo. But who cares, it will be fun.

[GlyphGryph]

You don't change past odds.

It's a 50% chance the coin came up tails (come on, we all know this one). It's a 2/3 chance that guessing tails will give you the correct answer.

But that is not because tails is more likely, but because you get to make more guesses if (and only if) you are right. The odds for the actual event clearly wouldn't change.

[anzki4]

About the whole coin-flipping thing; remember that the question was what would the sleeping beauty answer and that she wouldn't remember any time the coin has been tails, so eventually she would determine that there was some shady business going on and the coin-flip would always result to heads. After all it has been so every time in the past.

EDIT: Wait... So the amnesia-medicine would be administered when actually? Every Monday or only on Mondays when it's tails?

EDIT2: And also would she be told what was the result of the coin-flip that particular time?

[Sheb]

Wut?

[PanH]

2/3

Do it 100 times. There'll be 50 heads, and 50 tails, so 100 monday, and 50 tuesday.
50 M&H
50 M&T
50 T&T

Given the question, tails = 50 M&T + 50 T&T  /   150

[Loud Whispers]

Personally I haven't been able to understand how a plane lifts without generating any air flow under the wings. Unless the propeller has enough power to do it.

How the wings are designed, they create two paths for air to take; one above (longer, so more resistance), one below, less resistance. More air particles go the way of the short cut, creating lift for the plane. It doesn't generate this lift using its wheels.

It takes off.

Otherwise it's all about thrust.

[Blargityblarg]

Personally I haven't been able to understand how a plane lifts without generating any air flow under the wings. Unless the propeller has enough power to do it.

How the wings are designed, they create two paths for air to take; one above (longer, so more resistance), one below, less resistance. More air particles go the way of the short cut, creating lift for the plane. It doesn't generate this lift using its wheels.

It takes off.

Otherwise it's all about thrust.

Planes can fly upside down. (was that XKCD?)

[Aqizzar]

Personally I haven't been able to understand how a plane lifts without generating any air flow under the wings. Unless the propeller has enough power to do it.

How the wings are designed, they create two paths for air to take; one above (longer, so more resistance), one below, less resistance. More air particles go the way of the short cut, creating lift for the plane. It doesn't generate this lift using its wheels.

It takes off.

Otherwise it's all about thrust.

The oft-repeated and almost completely bunk Bernoulli Principle.  It's based on the idea that air traveling over the sloped upper plane of the wing has to travel faster to cover the same distance as air on the lower plane, or that air suddenly starts behaving like electricity and takes the path of least resistance, and either way creates more pressure under the wing and thus lifting it.

This is almost completely bunk because there is no universal law that says the air traveling over the top of the wing has to cover the distance in the same time as the under on the underside.  And because air is not electricity.

Wings generate lift by damming the air on the underside, creating a low pressure area behind the wing to pull air down faster over the top, creating low pressure there in turn.  These effects combined cause the wing to fly, and even then it all depends on the design of the plane.  As pointed out, a plane traveling with enough speed can fly upside down despite its wing curvature, but they sure can't fly without wings either.  It's all about the thrust and the control surfaces to direct where that lift air is going.