Sleeping Beauty

[Griffin]

I don't understand the paradox. Can someone explain it to me? It seems to me that no matter what day it is, it's a fifty percent chance of being heads. I figure there is a 25% chance of it being Tuesday(and being tails obviously), and a 25% of it being Monday and tails, and a 50% of it being Monday and heads. Thats fifty-fifty.

[mithrandir]

I think the paradox is that if it is Monday, there is 2:1 odds of it being heads (50%:25%), and if it's Tuesday it must be heads. So, 2/3 chance of heads on Monday, 0 on Tuesday, for an average of 1/3. But, I still think she would have to say 1/2.

[Quailman]

I'm not sure I follow your 2:1 odds of heads if it's Monday. At any given awakening, there are 3 situations with equal probability: Monday - Heads, Monday-Tails, and Tuesday-Tails. They are equal since Monday-H equals Monday-T, and if Monday-T happens, Tuesday-T will also occur. SB won't know which of these 3 outcomes is occurring upon awakening, so I think it's 67% that the toss would have been tails.

On the other hand, if Monday-T occurs, it is wiped out, making two possible outcomes, Monday-H and Tuesday-T, for 50-50, but I still think it's 2/3 Tails.

[mithrandir]

I couldn't figure out where the 1/3 came from until i read your post, and ayara's in the old forum. I was just speculating on what the paradox might be. I agree with the 1/3.

[Judge Phred]

I don’t see any paradox here. I feel that the question is slightly misleading, making it open to a bit of debate and seem like a paradox (an ambiguox?!) The difficulty is in the definition of what the statistical event is. Is it the tossing of the coin, or being woken to answer the question?

Here's my explanation:

Probability definition 1: Number of positive outcomes over total possible outcomes of the event. (The event is the coin tossing)

Of course, on waking up, you have no information other than that a coin was tossed once (the event)(assuming you don't cheat by checking your facial hair growth or something (yes I KNOW its a "she" in the problem!)) so the probability of it landing heads up is well known.
ANSWER: 1 in 2
REALITY CHECK: If you did this experiment every week for 2 years (ie: toss a coin 100 times), you would expect about 50 heads out of 100. (Even all thoses drugs and early mornings couldn't affect this!)

Definition 2: Fraction of times you would expect to be right if you picked a certain answer. (The event answering the question)

You are not actually being asked for an answer, only a probability of something which has already occurred. However, if you WERE asked to guess the answer, then would your answer on the first (and forgotten) day of the two when tails comes up count towards the total? If you DO count this, then you have an indeterminate number of events upon which you are basing your probability. As others have already shown, you have a 1 in 3 chance of being right if you call heads.
ANSWER: 1 in 3
REALITY CHECK: If you did this experiment every week for 2 years, (100 times), you would expect to be woken up 50 times on Monday with the coin on heads, and 50 times on Monday AND Tuesday with the coin on tails. Calling heads every time would yield 50 correct answers out of 150 total arousals (events)

CONCLUSION:
The way the question stands, I believe that the probability that the coin had come up heads is 1 in 2.

[Ghost Post]

My opinion would be the same as Judge Phred's. The problem is not so much in the problem but in how it's interpreted. The probability of a fair coin toss being heads is 50%. The probability of it being heads on a given awakening is 33 1/3% per awakening, but 50% per toss.

My head hurts!

[dethwing]

No matter how you dress it up, the anwser is still 1 in 2.
The fact that if it was tails she was asked the question twice has no relevence to the probabilty of the coin toss. Whether its monday or tuesday has no relevence to the probabilty either. Just because if it came up tails does not mean a thing, and anwsering a question twice does not alter probability either. For instance if i asked you what the odds of rolling a 6 sided die and getting a 6, and i said it would be a different day depending on what number i rolled, the answer is STILL 1 in 6 or 16.666666...% I rest my case

[Andy]

I answered 1/2 originally, but I have reconsidered and changed to 1/3. This is similar to a previously-posted puzzle about 10 black and white balls in a jar. The question being asked is not "What is the probability of a toss of a fair coin yielding a head?" (which is 1/2 according to my definition of "fair coin"), but "What is the probability that the fair coin we tossed Sunday night came up heads?" The question is asked of S.B. upon her awakening, so the event in question is a combination of the outcome of the coin toss, and the awakening of S.B. In this set of events, the coin comes up heads 1/3 of the time, since each coin toss yielding tails produces two events.

[Ghost Post]

Sleeping Beauty should answer "Heads" one out of every three times she is awoken.

This statement answers the question given, which is "Would we answer 1 in 2, or 1 in 3?"

I would answer 1 in 3, and this is why:

The coin is tossed once per week, on Sunday evening, at the same time she is put to sleep.

In the event that it is Heads, she is woken Monday morning and asked.

In the event that it is Tails, she is woken Monday morning AND Tuesday morning. However, the coin is NOT flipped again.

Imagine it this way: (and you can try it at home)

Make two columns- label one "Heads", label the other "Tails"

Flip a coin as many times as you'd like. Each time it's heads, mark one tally in the Heads column. Each time it's Tails, mark one tally in the Tails column, FORGET YOU DID THIS, and do it again, for a total of two tallies per Tails toss.

As the number of trials increases, the numbers would settle down into 1/3 Heads, 2/3 Tails, and you'd have, in actuality, 150% the number of tallies as the number of flips.

Thus, S.B., for the best odds, and knowing the situation, should answer Heads 1 out of 3 times she is woken.

[Ghost Post]

If we make this a betting game instead of an experiment, it is easy to determine S.B.'s best strategy.

Suppose that each time S.B. guesses right, she wins $1, and each time she guesses wrong, she loses $1. She has three strategies she can follow: always guess heads, always guess tails, or randomly guess either heads or tails. (A fourth strategy, to alternate heads and tails, is impractical because of the amnesia).

If the strategy is to always say heads, there are 2 possible outcomes:
1.Flip result = Heads
Monday guess = Heads
Tuesday guess = n/a
Payoff = $1
Probability = .5
Expected payoff (Payoff x Probability) = $0.50

2.Flip result = Tails
Monday guess = Heads
Tuesday guess = Heads
Payoff = $-2
Probability = .5
Expected payoff = $-1

Total expected payoff = $-0.50 (Sum of possible expected payoffs)

If the strategy is to always say tails, there are 2 possible outcomes:
1.Flip result = Heads
Monday guess = Tails
Tuesday guess = n/a
Payoff = $-1
Probability = .5
Expected payoff = $-0.50

2.Flip result = Tails
Monday guess = Tails
Tuesday guess = Tails
Payoff = $2
Probability = .5
Expected payoff = $1

Total expected payoff = $0.50

If the strategy is to guess randomly, there are six possible outcomes, and the total expected payoff is $0. (The calculation of the total expected outcome is left as an exercise )

Therefore, S.B.'s best strategy is to always guess tails. If you change the game so she wins $1 for each right answer, but loses nothing for each wrong answer, the total expected payoffs change, but the "Tails" strategy is still the best.

I know this doesn't address the probability of the flip, but I thought it was a good start.

[extropalopakettle]

It's 1/2.

Suppose that after the experiment is over, she will know whether she received the amnestic drug or not (she's mentally keeping track of days, and will compare the actual day, when she knows it, with the apparent day, and if they differ, she'll know she was given the drug).

When they ask her the question each morning, she asks herself "What is the probability that when the experiment is finally over, I will discover that I received the drug?"

I'd think it's pretty intuitive that she must answer 1/2 to that question (she knows that if the experiment is repeated, about 1/2 of the time she will have gotten the drug).

But she receives the drug precisely if the coin landed tails, so the probability of that must also be 1/2.

[ckalish]

Another twist: If we assume that in the case of tails, SB will be awakened at 6:00A.M. and asked the probability of heads every morning to infinity and, of course, given the amnestic every evening, then it logically follows that her answer must be "zero probability of heads". Well, if .9999999999999...to infinity equals 1, then why not?

[araya]

--When they ask her the question each morning, she asks herself "What is the probability --that when the experiment is finally over, I will discover that I received the drug?"

The answer to this question on Sunday evening is 1/2.. The answer to this question after they tell her 'the experiment is now over' is 1/2. The answer to this question when they wake her up and ask is 2/3.

I don't know, if you've thought about the problem for this long and still insist on 1/2 as the correct answer, I guess there's nothing I can say to change your mind. Let's wait and see what the Minotaur has to say. I keep trying to prove the 1/3 answer based on conditional probability rules, but I can't do so properly. I also can't prove the 1/2 answer based on any allowable probability rules. However, I think the arguments I posted in the other forum were very strong in favor of the 1/3 answer.

ckalish: a lot of paradoxical results arise when you start dealing with infinity. For instance, the number of even numbers is equal to the number of all numbers. For every number n, there is an even number 2n, with a one to one correspondance. But how can this be? From any interval from 1 to an even number, we know that there are twice as many numbers as even numbers. It is true because there is no limit to infinity. For your example, the probability is indeed zero of heads.

[Ghost Post]

Araya, I appologize in advance if I am just being stubborn, but I have to disagree with you.

Lets discuss the infinite problem first. I disagree with your answer of 0 probability for heads. Put yourself in the experiment. You go to sleep on Sunday night. You then regain consciousness after a night's sleep. You are then asked what the probability was that the coin had come up heads. You think as hard as you can but can't remember if this same exact thing happened yesterday. Do you answer that there is absolutely no (1 in infinity) chance of heads? NO. There is a 50% chance that it is Monday and the experiment is over. There is also a 50% chance that it is any given day from Monday to infinity. Therefore you must answer 1/2.

The same can be said of the given problem of Monday and Tuesday. There is a 50% probability that it's Monday and the coin landed heads, and there is a 50% probability that it's Monday or Tuesday and the coin landed tails.

Conditional Probability plays no part in this experiment. There is a 50% chance of one thing and a 50% chance of another (whether you are talking about HEADS/TAILS or MONDAY/MONDAY&TUESDAY). Please explain where my logic is faulty. I don't see how the act of being awoke can change the probablility of the result of the toss since the result of the toss doesn't change the probability of the outcome of the experiment. Do you disagree that there is a 50% chance of it being Monday with heads and a 50% chance of it being either Monday or Tuesday with tails? On awakening, there is a 50% chance that you were asked the question yesterday and a 50% chance that you weren't.

1/2

[Ghost Post]

note: i edited this message completely, as i added nothing new before. i do have something new to add now, however.

dorcas: imagine this problem is done a million times.

i'm going to take the paradox out of it, then pull it back in. please follow me:

no paradox: SB goes to sleep, coin is flipped. in the morn, she is woken, and asked if the coin flipped heads. this happens for one million weeks. it is obvious that, to be completely right, SB shoud answer heads .5 million times.

with paradox: SB goes to sleep, coin is flipped. in the morn, she is woken, asked if the coin flipped heads.

let's assume that for the million times the coin is flipped, exactly one half of the times it lands heads, and exactly one half of a million times it's flipped tails.

for each time that it's flipped tails, SB gets asked TWICE.

therefore, SB gets asked the question 1.5 million times. to be right all the time, she will answer .5 million times heads, and .5 + .5 million times tails.

therefore, to be completely right, she should answer heads one out of three times she's asked.

the kicker is that she doens't know how many times she's been asked- but it settles down to 150% the number of weeks this happens.

where did i go wrong?
--Scotty

[This message has been edited by Scotty (edited 10-04-1999).]

[This message has been edited by Scotty (edited 10-04-1999).]

[Ghost Post]

She is not asked to answer "heads" or "tails" and she has no interest in being right. She is simply asked "what the probability was that the coin had come up heads." Using any form of the statement "she should say tails because she'd be right two out of three times" does not prove the probability of heads being 1/3. Araya has made the best arguement for 1/3 but I am not convinced on the basis that the act of waking up does not effect the result of the toss. Yes you are asked the question twice as many times with tails but that still only happens 50% of the time. How does the fact that they ask you the question twice change the probability of the result of the toss? Once again, upon awakening there is a 50% chance that you were asked the question yesterday (tails) and a 50% chance that you weren't (heads).



[This message has been edited by Dorcas (edited 10-04-1999).]

[extropalopakettle]

Araya replied to:
When they ask her the question each morning, she asks herself "What is the probability that when the experiment is finally over, I will discover that I received the drug?"

with the following:
The answer to this question on Sunday evening is 1/2.. The answer to this question after
they tell her 'the experiment is now over' is 1/2. The answer to this question when they
wake her up and ask is 2/3.

Why can't she figure that out?

If she knows ahead of time (as you do) that when they finally tell her "the experiment is now over" (at which point she will have at least as much information as she did at any other time), that she will then answer 1/2 (to the above question), why would she at any time give a different answer?

You're saying she can think the following when they wake her:
1) When they tell me the experiment is over, I will then know there is a 1/2 probability that I have received the amnestic drug.
2) They will definitely tell me at some point that the experiment is over.
3) Therefore, I will definitely know later that there is a 1/2 probability that I have received the drug.
4) There is a 2/3 probability that by the time the experiment is over, I will have received the amnestic drug.

In other words, you're saying it is possible to know that you will later know the answer is 1/2, without now knowing the answer is 1/2.

That's like saying "I don't know what Sleeping Beauty's name is, but tomorrow I will know that her name is Jane".

[Ghost Post]

Thank you for clearing that up. I was wrong in saying that there is a 50% chance that you were asked the question yesterday and a 50% chance that you weren't.

The statement really should have said that there is a 50% chance that you weren't asked the question yesterday and the experiment is over, and there is also a 50% chance that either you were asked the question yesterday and the experiment is over or you will have to go thorough this whole process again. This still doesn't change the probability that the coin had come up heads.

[dethwing]

I dont know about anyone else, but im gonna be happy when the finally post the anwser all these 1/3s , 2/3s , 3/4 , 1/x , or 7/9 are starting to iritate me
Flip a coin.
Heads:Heads
Tails:Tails

Whats so hard about that?

[araya]

ok, extropalopakettle, (you really should shorten that, it takes forever to write ;-)

On Sunday evening, she knows the situation exactly - as of that moment, the probability is 1/2 to the best of her knowledge. When the experiment ends, she is in the same situation as on Sunday, really, assuming that she can't tell that it's wednesday or tuesday, and she doesn't have any idea whether she got the drug or not. When they say to her, 'the experiment is now over', say, right after she answered *the question*, she knows it's either monday heads or tuesday tails, so 1/2. But, when she is asked *the question*, the uncertainty as to what day it is causes her to take into account all the information regarding the experiment conditions, etc.. and give a different answer, because her exact situation is not known to her. When I said she could prepare her answer on Sunday evening, what I meant was not that 1/3 would be her answer at the time, but she could write it on a piece of paper in preparation for being woken up. At the time her answer would clearly be 1/2, because she knows her situation exactly.

---That's like saying "I don't know what Sleeping Beauty's name is, but tomorrow I will know that her name is Jane".

Well, not really.. you ARE given new information when they say 'the experiment is over', because you know it's not monday if it was tails.

Dorcas: you're not being any more stubborn than I am - we all want to make our viewpoint the correct one. In response to your question - 'Please explain where my logic is faulty', I will reiterate my main point. The people who believe the answer is 1/2 seem to agree that this arises because there is a 50% chance of Monday/heads occurring and a 50% chance of Monday/tails and Tuesday/tails occurring. But you're totally leaving out the partner to Tuesday/tails, namely Tuesday/heads. You say 'but it can't be tuesday/heads, because you will be woken up to answer the question, and that doesn't happen on tuesday/heads'. This is what conditional probability does - you take into account the fact that you have been woken up, and compare with the total chance of being woken up, in order to find a better estimate of the event in question. You have to include the situations where something doesn't happen in your analysis, even though you know it has happened.

[Ghost Post]

DORCAS WROTE:
Once again, upon awakening there is a 50% chance that you were asked the question yesterday (tails) and a 50% chance that you weren't (heads).

I think that there is a flaw in that sentance, and it should read:

Once again, upon awakening there is a 50% chance that you were asked the question yesterday (tails) and a 50% chance that you weren't (heads OR tails).

See the signifigance?

Nick

[mithrandir]

Yes there is a difference, but according to that sentence the probability should be 1/4, 75% tails, 25% heads. There is actually a 33(.33333...)% chance that IF you are woken up and asked the question, you were woken up and asked the question yesterday(tails), and a 66(.66666...)% chance you weren't(heads or tails).

[Murray]

I've tried to keep my silence on this one because it's supposed to be a paradox, i.e. there is no correct answer. I think Judge Phred was right to say this is an "ambiguox." When I first read the question it took me a while to figure out what the paradox was. I thought, "She should say 1/3, obviously." But the question preys on the ambiguity of language and seems also to ask, "Just answer me, when I flip a coin, what are the odds of heads?" Unfortunately, this interpretation of the question is simply not correct. If you read the question in this way, you are ignoring the added information that you have, namely the conditions of the experiment combined with the knowledge that you are being awakened. I can't help feeling that if you answer 1/2 to this question, you would probably not change doors when Monty Hall opened up a loser.

Consider this new experiment. I hope it will offer something more concrete to you all.

Most of the same conditions apply, but instead of being drugged and awakened again on Tuesday, things just go on as normal and the experiment is performed again the next week. BUT, if the (fair) toss comes up tails then they use a different coin the next week--a coin with tails on both sides. Then the next week they go back to using the fair coin. If the original toss comes up heads, then they just flip the fair coin again the next week.

At the end of 150 trials, there have been 50 heads and 100 tails. Fifty of the tails tosses were dependent on the other 50. If you do not agree that this experiment will yield the EXACT same results as the original, then you might as well stop reading this. I could have left the drugging and Tuesday arousals in and not changed my experiment. I merely changed these things because they cause confusion. Suffice it to say that the results, methodology and appearance to the test subject are all identical. I'm simply substituting a two-tailed coin for the guaranteed-Tuesday-tails of the original experiment.

Now, let's take another look at the results: We flipped a fair coin one-hundred times and we flipped a two-tailed coin 50 times. So on any given night the experiment can be simulated by saying, "S.B., we are going to flip a coin tonight. It will be a fair coin with probability 2/3 and it will be a two-tailed coin with probability 1/3. When we wake you in the morning, please tell us the probability it was heads."

S.B. thinks, "Heads will never come up on the two-tailed coin, and the other coin, which will be used 2/3 of the time, will come up heads with probability 1/2. Half of 2/3 is 1/3. So the answer is 1/3."

I can't think of a more concrete way of putting it.

[Ghost Post]

I was in the middle of typing my response when you posted your reply, Murry. This was written in response to mithrandir, but it can also be applied to your argument. I personally don’t believe that you can separate Monday tails and Tuesday tails. Please read on for an explanation.


My initial statement was wrong; read my reply after extropalopakettle. But I still don't agree with you, mithrandir. There is actually a 50(.500000...)% chance that IF you are woken up and asked the question, that the coin landed heads and the experiment is over, and a 50(.50000...)% that the coin landed tails and the experiment is either over or you will be awaken tomorrow. Now if you were asked "what is the probability that the experiment is over?" or "what is the probability that it is Monday?" or "if you were to place a bet would it be on heads or tails?" or something of the like, the 1 in 3 probability comes into play. However, with the posed question "what the probability was that the coin came up heads", the only answer is 1/2. The act of awakening does not give you any new information to change your answer from 1/2. Even if you knew what day it was your answer would still be either 1/2 on Monday or 0 on Tuesday. These answers cannot be combined to make 1/3. It appears to me that the 1/3 "holdouts" are answering the question "what is the probability that the coin came up heads AND the experiment is over." The experiment being over is irrelevant to the probability that the coin came up heads (since you have no idea what day it is or if the experiment is over). Now some of you will say that if the coin landed heads the experiment is over. True, but the chance of that happening is still 50%. Furthermore I don't see how you can separate Monday/Tuesday Tails. There is no probability involved in Tuesday tails outside of the 50/50 coin toss. Tuesday automatically follows Monday 100% of the time for tails. There is a 50% chance that you will be woken up once and a 50% chance that you will be woken up twice. In your argument, Murry, you change the experiment. You are using two different coin tosses (even though one is fake). The original question is asked of a single, fair coin toss. What you are saying Murry is that if this experiment were ongoing, each week a coin was flipped. However, if a toss came up tails, the next weeks “fair” coin toss would be put off another week and the falsified toss would be made. Reduce you problem to one coin toss and you can see the difference. There is a 50% chance that the falsified coin will be flipped and a 50% chance that it won’t. The result of the falsified coin does not play into the original probability of a coin toss. Some people use the argument including a mythical “Tuesday heads”. There is no such thing. The problem can be just as easily stated that if the coin lands heads we will wake you on Monday and ask the question, and if the coin lands tails we will wake you on Tuesday and ask the question give you the mind eraser and wake you again on Wednesday and ask you. It really makes no difference. There is a 50% probability of one thing happening and a 50% probability of another. Say that there was something wrong with the drug that they gave you on Monday evening and you croaked. Oh no, you were never asked the question on Tuesday. Well, it doesn’t matter because “the probability that the coin came up heads” is 1/2. Once again I apologize for being a little stubborn. I have tried to convince myself of the 1/3 argument. I would love to believe the answer to be 1/3. That would make it a real paradox, having to use logical thinking to come up with an illogical answer. But I cannot do it. Every time I try I end up proving 1/2.




[This message has been edited by Dorcas (edited 10-05-1999).]

[Quailman]

I think those of us who support the 1/3 probability are reading the implied portion of the question: What is the probability that the coin landed heads (and we woke you up)?

In my opinion, it makes a difference that SB has been awakened, and it affects the probability.

Someone asked on another thread how often new puzzles are posted. Why should the Minotaur post a new one when he is getting so much mileage out of this one? What is the probability that the debate will start all over again once he posts the "solution." Better yet, what is the probability that there is a solution?

[Murray]

Quailman, I would say the probablility of the argumant starting all over again is 1/2, and the probability that there is a solution is also 1/2. Why can I state these things with such confidence? Because I'm the kind of person who will answer 1/2 to all questions despite all the evidence to the contrary. I even think the answer to the Monty Hall question is 1/2.

Now, I must pose a question to all the -actual- 1/2ers out there. It seems that all of your arguments hinge on the simple fact that a coin toss will come up heads half the time and tails half the time. I think we can all agree with that.

So, what is the point of the rest of the problem, drugging our poor Sleeping Beauty and waking her at an ungodly hour? What is the point of all that garbage if all we're doing is flipping a coin? Why isn't the question phrased, "What is the chance that a fair coin toss will come up heads?" The inventor of this problem was just a little too verbose, and all this additional information is simply to be ignored?

[This message has been edited by Murray (edited 10-05-1999).]

[Ghost Post]

To all the 1/3's, it seems that all of your arguments hinge on changing the problem or making assumptions about the experiment that you don't know and can't prove are valid.

A-ha, now we are getting somewhere. Are we supposed to read anything into the problem? Are we to assume that there is something implied in the problem? S.B. was not told that anything was implied. I think we have to take the problem as is without reading anything into it. As the question stands (what is the probability that the coin came up heads), the answer I would give is 1/2. However, if a different question was asked to me, I might change my answer. I believe that you are supplied all the information needed to make an answer. I don't believe that you have to read anything into the question. In a valid experiment there are no implications. In something like engineering, there are assumptions, but the answers are only valid as long as the assumptions are valid. How do you know that your question with implications is valid? You are not the one asking the question. Take the puzzle for what it is and give an answer. I keep coming up with 1/2.

Don't get an attitude here Murry. The Monty Hall puzzle can be proven with facts. In proving 1/3 as an answer for the Sleeping Beauty problem, although you MIGHT be using logical assumptions they are still that - assumptions.

And we all know what happens when you assume.

[This message has been edited by Dorcas (edited 10-05-1999).]

[This message has been edited by Dorcas (edited 10-05-1999).]

[This message has been edited by Dorcas (edited 10-05-1999).]

[Murray]

I'm not talking about the implications for S.B. There are no implications for her. She is given the situation in its entirety and plays the game.

The implications I'm talking about are those for the puzzle solvers--you and me. I hate to belabor this point, because it is so far off-topic that we should probably talk about it in the off-topic section, and because we should be addressing the actual problem, not it's implications. I'm actually sorry I brought it up. I'll rephrase my sentiments and then drop this topic, because I'd rather see someone explain to me the flaw in my "fair coin and two-tailed coin" version of the problem, as I am not convinced that it can be shown to be any different from the original problem. So here is my stupid, off-topic, meant-to-be-rhetorical question rephrased and best left unanswered:

Why is this experiment so complicated if the answer is so trivial?

Now back to the real question...

Sleeping Beauty

_________


For the record, "attitude" is not to be confused with tongue-in-cheek. For my part, I think that this problem is actually a paradox--1/2 and 1/3 are both neither right nor wrong--but I think that this paradox is based upon a deficiency of the English language not being able to specify whether the question is supposed to mean, "We flipped a fair coin: what are the odds that it landed heads-side-up?" or, "We flipped a fair coin and left it there: what odds would you use for wagering purposes that the heads side is still up?" The answers should be 1/2 and 1/3 respectively.

[This message has been edited by Murray (edited 10-05-1999).]

[Ghost Post]

I was thinking about the problem and I think I came up with the crux of the matter, if you will.

Does the word probability imply a future event or can it be extended to be a past event?

imagine a scenario wherein i have a coin. i ask you "what is the probability that (when i flip it) this coin will come up heads?" every sane person would say "1/2". if i then flip the coin, and it lands heads, and you see that it landed heads, and I ask you "What is the probability that the coin landed heads?"

would you say "1/2, because it's a fair coin"? or "1, because i saw that it landed heads and there's no other option for it to have landed differently."

agreed, that if i asked you "what WAS the probability that the coin landed heads?" you'd say "1/2". but now, after the event, the probability only remains 1/2 if you know nothing about the event.

if, when asked "what IS the probability that it landed heads?" you'd STILL say 1/2, then the answer to SB should be 1/2. if you'd say "1", your answer to SB should be 1/3.

This is a new thought direction for me, so I don't know which I believe to be true. I also don't know the true, accurate, mathematical definition of 'probability'.

What are everyone else's thoughts? Is this on track or am I completely wrong?

[Griffin]

The problem reads:

"(She would be) asked what the probability was that the coin had come up heads."

Personally, after reading the many posts on the subject, here is what I think it means:

Pretend that the people who are conducting the experiment just left the coin sitting on the floor after they had fliped it. The question then reads "What is the probability that rigth now the coin is laying on the floor heads up?" The obvious answer becomes 1/3. But I'm still not sure.

[Ghost Post]

So, it looks like we've boiled the paradox down to english.

the same paradox can be suggested by the following scenario:

Today is Thursday.

Miss. Brown married Mr. White on noon at Tuesday, thus becoming Mrs. White.

On Wednesday, Mr. Green, who lives next door, was asked what his female neighbor's name was.

What would he answer?

He would answer Brown if he felt he got asked what her name WAS. He would answer White if he felt he got asked what her NAME was.

Maybe this stretches it, but the bottom line is we don't know the time to which the questioners refer when SB was asked. We also don't know the exact question: "What was the probability?" or "What is the probability?"

Does this help? It's obvious, given the paradoxical nature of the question, that there is no right answer.

but i still say 1/3, because its fun to say that the probability of a coin landing heads is 1/3, rather than the probability WAS 1/2.

[Ghost Post]

I support the claim that the probability of Heads is 1 / 3, but even those who agree with me, I feel, should consider my reasoning.

It is obvious that, upon awakening/questioning, it is either Monday or Tuesday, and the coin is either Heads or Tails. Therefore, let the sample space be partitioned into four disjoint sets:

• Monday/Heads
  
• Monday/Tails
  
• Tuesday/Heads
  
• Tuesday/Tails
  

Since no other possibilities are apparent to me,

P(Mon & Heads) + P(Mon & Tails) + P(Tue & Heads) + P(Tues & Tails) = 1.

Since the subject will not be awakened/questioned on Tuesday if the coin flip results in Heads, I hope we all agree that the event Tuesday/Heads is impossible. Therefore,

P(Tue & Heads) = 0.

Now, let us consider what happens when we hold one variable constant and examine the other.

Assume that it is Monday. If it is Monday, the conditional probability of Heads is equal to the conditional probability of Tails, since the coin is fair. Every flip of Heads results in exactly one Monday interrogation, and every flip of Tails results in exactly one Monday interrogation. (In multiple runs of the experiment with a fair coin, we would expect approximately the same number of Monday/Heads and Monday/Tails events.) Therefore,

P(Mon & Heads) = P(Mon & Tails).

Assume that the coin flip results in Tails. If Tails occurs, there will be exactly two awakenings, one on Monday and one on Tuesday. Every Monday/Tails guarantees a Tuesday/Tails event, and every Tuesday/Tails event implies that a Monday/Tails event has occurred. (In multiple runs of the experiment there will be exactly the same number of Monday/Tails and Tuesday/Tails events.) Therefore, if the coin flip results in Tails, it is intuitively obvious that Monday and Tuesday awakenings are equally probable. Therefore,

P(Mon & Tails) = P(Tues & Tails).

Then by the transitivity property of equality,

P(Mon & Heads) = P(Mon & Tails) = P(Tues & Tails).

That is to say, that since Monday/Heads and Monday/Tails have the same probability and Monday/Tails and Tuesday/Tails have the same probability, then all three events have the same probability.

Remember that

P(Mon & Heads) + P(Mon & Tails) + P(Tue & Heads) + P(Tues & Tails) = 1.

Then, since P(Tue & Heads) = 0, we conclude that

P(Mon & Heads) + P(Mon & Tails) + P(Tues & Tails) = 1.

However, since all three of these events are equally likely, we can conclude with certainty that

P(Mon & Heads) = 1 / 3

Since we know that Heads flips result only in Monday awakenings, and never Tuesday, we therefore know that

P(Heads) = 1 / 3

upon the subject's awakening, given the circumstances of the problem.


[This message has been edited by Derkage (edited 10-05-1999).]

[This message has been edited by Derkage (edited 10-05-1999).]

[dethwing]

If Tuesday Tails = Monday tails, shouldnt they cancel eachother out? since tails is tails no matter what day it is? On a side note, my dad agrees 1/3 but for a compltetly different reason. He claims that the odds of tails is just plain better than heads. Is there and truth to that?

[mithrandir]

This is crazy, I posted my two cents this morning, and already there are 11 more posts!

Anyway...

I don't think we are accomplishing anything discussing this much anymore, I and several others are convinced of 1/3, and the rest are just as firmly convinced of 1/2. It rather impresses how many variations people have come up with to prove their points, but I think we are running out of ideas. I just hope the "solution" (if there is one) is posted soon.

[araya]

Agreed, I'm very tired of talking about this 'paradox'. I want to see what the minotaur says about it, although I doubt that it will satisfy everybody. Notice that the first post I wrote on this topic (old forum) stated that the paradox lies largely in the wording of the question. Nevertheless, I have to believe that the question 'what was the probability that the coin toss was heads', taken to totally ignore all of the experiment conditions and focus on the simple 1/2 probability, is just plain stupid and pointless. Anybody who focuses on the unfortunate use of 'was' for 'is' misses the point of the whole question.

[Ghost Post]

Just a comment: the use of the words, "had come up" verfies that she is asked about the outcome of a past event, not the probable outcome of a future even.

Consider,
"What is the possibility that the coin had come up heads?"

"What was the possibility that the coin had come up heads?"

The past tense word "had" must be changed to "would" in order for the odds to be placed before the event.

Let me use a different event.

"What is the possibility that I had come home before six?"

"What was the possibility that I had come home before six?"

Then, "What was the possibility that I would come home before six?" This question asks what the odds were before the event occured.

Now, as far as the answer goes I have a couple of thoughts. First, lets assume that the question is "What are the odds that the result of the coin toss was heads."

Next, we have four questions to answer. What are the odds that today is Monday, that today is Tuesday and for each day, what are the odds that the toss had come up heads?

Well, the odds of it being Monday are 2/3, and the odds of Tuesday are 1/3. Why? I'll write a post script.

Now, the odds of heads on Monday are 1/2, and the odds of heads on Tuesday are 0/1.

Ok then. We wake up. There is a 66% chance that it's Monday, half of that that it's Monday and Heads. That gives us 33% towards Heads. There's also a 33% chance that it's Tuesday with a 0% chance that it's heads, which doesn't change a thing. So, we can confidently answer that there is a 1:3 chance that the coin toss had come up heads.

Nick

P.S. Now, why is there a 1/3 chance that today is Tuesday? There will always be a Monday, and there will be one Tuesday for every two Mondays. That actually gives us a 1:1.5 chance of it being Monday, since we only have a 50% chance of there being a Tuesday. However, 1/3 is the language of the puzzle so...

[mwf]

I have to say it would be 1/2 for toss being heads.

When I woke up that moring I would have to ask my self if this is Monday or Tuesday. If today is Monday then there is 1/2 chance that the coin landed heads. If today is Tuesday then the chance of the coin had to land tails, but there is a 1/2 chance that I'm wrong and to day is Monday and the toss is not tails.

Now there is a 2/3 chance that they will bring me my coat and I can go home and find out what day it is. Then there is the 1/3 chance that they will bring me the needle and I will now that it is Monday and then soon forget about it.

Last of all the question has no right answer. The question was "Would we answer 1 in 2, or 1 in 3?". It did not ask what is the right answer, only how we would answer the question. So 1 in 2 is just as good a answer as 1 in 3.

Well I thought I would give my 2 cents worth or was that coin flip worth.



[This message has been edited by mwf (edited 10-06-1999).]

[Ghost Post]

Can anyone explain mwf's statement:

"If today is Tuesday then the chance of the coin had to land tails, but there is a 1/2 chance that I'm wrong and to day is Monday and the toss is not tails."

I agree, that if today is Tuesday, then the coin had to have landed on Tails. However, if today is Tuesday, then there is no chance that one is wrong in claiming that it is Tuesday, and there is absolutely no chance that it is Monday, or the coin is Heads. Where does he get the "1/2 chance"?

However, upon awakening, there is a 1 / 3 probability of it being Tuesday, and a 2 / 3 probability of it being Monday.

Also, what relevance do the following statements have to the problem?

"Now there is a 2/3 chance that they will bring me my coat and I can go home and find out what day it is. Then there is the 1/3 chance that they will bring me the needle and I will [k]now that it is Monday and then soon forget about it."

[mwf]

Derkage this is the only relevent statment I made.

Last of all the question has no right answer. The question was "Would we answer 1 in 2, or 1 in 3?". It did not ask what is the right answer, only how we would answer the question. So 1 in 2 is just as good a answer as 1 in 3.

The rest is only how I would answer the question if were being tested in the experiment. Except for the part about the coat and needle. That was me just have some fun with some of the earlier posts.

Remember this is just me opinion and why I would answer 1 in 2.

The only right answer for me is the one I give and the only right answer for any one that was being tested is the one that they gave. The people doing the study are probably only going to speculate on the collected data and maybe publish the results.

Now back to my 1/2 chance Tuesday is Monday.

Yes, there is a 2/3 chance that when I wake up it is Monday. If it is Monday as we have all agreed in the prior posts the coin toss is 50/50 or we would answer 1 in 2 for heads. Now as the problem stated half the time we go home and the other half we will forget that Monday never happened and wake up Tuesday. So as I see it 1/2 of the time I will wake up Monday and go home and 1/2 of the time wake up on Monday only to forget Monday and wake up Tuesday and go home. So if I know that it is Tuesday then the coin toss must have been tails and we can all agree on that as well. The problem is I have no way of knowing it is Monday or Tuesday until I leave the experiment and check the date. When I wake up I will have to ask my self if it is Monday or Tuesday. I say it Monday the coin toss is 1 in 2 heads and half the time I will go home and the other half wake up on Tuesday. Having forgot Monday. So if I were to make the ASSUMPTION that today was Tueday when I woke up. I would be WRONG half the time and the coin toss would had been heads.

I'm sure that if you were to write your self a progam to test this whole problem, you only answer head or tails correct 50% of the time.

[Ghost Post]

Toss: Heads Wake: Monday-Heads (1)

Toss: Tails Wake: Monday Tails (1)
Wake: Tuesday Tails (2)

Toss: Heads Wake: Monday Heads (2)

Toss: Tails Wake: Monday Tails (3)
Wake: Tuesday Tails (4)

Toss: Tails Wake: Monday Tails (5)
Wake: Tuesday Tails (6)

Toss: Heads Wake: Monday Heads (3)

Toss: Tails Wake: Monday Tails (7)
Wake: Tuesday Tails (8)

Toss: Heads Wake: Monday Heads (4)

Toss: Tails Wake: Monday Tails (9)
Wake: Tuesday Tails (10)

Toss: Heads Wake: Monday Heads (5)

Toss: Heads Wake: Monday Heads (6)

Toss: Tails Wake: Monday Tails (11)
Wake: Tuesday Tails (12)

Total Weeks: 12
Total Heads: 6
Total Wakings: 18

Heads / Wakings = 1/3

So, what were the odds, at any given waking, that the coin had come up heads? 1/3

Most people seem to agree to this. The question in their minds is what was the intend of the quesiton. I posted on that earlier.

Is the question asking, what are the odds that a fair coin toss will land heads up. Or, is the question asking, what are the odds that this particular fair coin toss came up heads.

Honestly, I don't think that the language is ambiguous in any way. I think that this is a more straight forward, answerable, and understandable question than Monty Hall.

By the way, I'm one of the Odds are better to switch crowd (counter intuitive, isn't it?)

I hope we're all having fun. It's what we're here for (unless it's about pride and self worth).

Shrug,

Nick