I'm having trouble finding the paradox here. As with many probability puzzles, the problem seems to be a hidden conditional in the probabilities.<p>There's no change in belief happening, only two different beliefs. First is "1/2 of the times it is flipped, the coin will be heads". Seconds is "1/3 of the times I am awakened, the coin will be heads". The coin flip is fair, but the decision to ask the question is biased.<p>This is extremely simple to demonstrate by resorting to the absurd case. If we change the awakening ratio away from 1-2, the absurdity of saying "1/2" becomes increasingly clear. At 1-9, tails will be the correct guess 90% of the time the question is asked. At 0-1, heads isn't even a possible outcome on awakening.
Can someone explain the half position clearly? I don't understand how anyone could think half is the correct answer.<p>Let me frame the question a different way. You are one of three volunteers in separate rooms. I flip a coin and if it's heads I ask one volunteer (at random) to guess the outcome. If I flip tails I ask two of the volunteers (again, at random) to guess the outcome.<p>You know the rules I will follow, but you cannot tell if anyone else has been asked before you. I open the door and ask you to guess the outcome of the flip. What do you guess?
I don't see the problem in their problem:<p>> Whenever SB awakens, she has learned absolutely nothing she did not know Sunday night. What rational argument can she give, then, for stating that her belief in heads is now one-third and not one-half?<p>The argument that she can give is clear: she <i>might</i> have learned something, but the memory was taken from her. And she knew that they would take the memory from her. So yes, she hasn't learned anything new, but that's a cop-out—the problem explicitly prevents it.<p>I mean, from that perspective, she doesn't even need to go to sleep! They could ask her, "when you wake up, how likely will you think that the coin was heads?" and get the same (correct) response of 1/3. That result is not based on her "learning new information" (since the situation forbids it), it's based purely on the situation as described.
Here is a surefire way to win the lottery.<p>Start by picking numbers. Your chances of winning naturally are very small, so I will make an arrangement with you. You go to sleep before the winning numbers are read. Afterwards, if you have won, I will wake you up N times, administering our trusty forget potion. But if you have not won, I will wake you up only once.<p>As we have established in this thread, by increasing N, your chances of having guessed the lotto numbers correctly upon awakening approach 100%.<p>I wake you up, you apply Bayes' Theorem, and then rejoice, for you almost certainly have won the lottery!
"[…] to what degree should you believe that the outcome of the coin toss was Heads?" is a terribly unrigorous way to phrase this question. I suspect it's where all the confusion sets in.<p>Are we adjusting the natural prior of a coin toss based on information we now have? We have no information; 50%.<p>Are you placing a bet each time you are awake? You get to bet <i>twice</i> if the coin comes up Tails; 2:1 (33%) odds against Heads balances the tables.<p>Are we judging how often we'd be correct if we guessed Heads every time we were awoken? Depends on the number of trials. With only 1 trial; 50%. With 2 trials; 42%. With infinite trials; 33%.<p>Nothing begets a paradox like an ill-posed question.
A similar problem is God's Coin Toss as described in Scott Aaronson's excellent lecture series (and book) "Quantum Computing Since Democritus":<p><a href="http://www.scottaaronson.com/democritus/lec17.html" rel="nofollow">http://www.scottaaronson.com/democritus/lec17.html</a>
Not a paradox...here is a proof that it is 1/3<p>Let P(T) = probability that coin landed on tails, P(H) = probability that coin landed on heads.<p>Let "1st" denote the event that it is the first time you are awaken, "2nd" the second time.<p>Note that P(T|1st) = 1/2 (if you were told that this is the first time you were woken up, it's equally likely that the coin landed heads or tails). And of course, P(T|2nd) = 1<p>By the definition of conditional probability,
1/2 = P(T|1st) = P(1st|T)<i>P(T)/P(1st) = (1/2)</i>P(T)/P(1st)
Hence P(T) = P(1st) = P(T)<i>1/2 + P(H) = P(T)</i>1/2+(1-P(T)) = 1 - P(T)/2
-> P(T) = 2/3
Both are right: the two camps posit completely different things.<p>Halvers stipulate a single experiment. Thirders stipulate infinite experiments.<p>It is pretty straightforward math to show these are not inconsistent, and there are even options in between. <a href="https://stats.stackexchange.com/questions/41208/the-sleeping-beauty-paradox/169582#169582" rel="nofollow">https://stats.stackexchange.com/questions/41208/the-sleeping...</a>
From my understanding, the answer is 1/2.<p>Let us assert that the probability of the coin flip being heads is 1/2.<p>Now, you have awoken. Regardless of if you've awoken to the Heads flip, or the first time to the Tails flip, or the second time to the Tails flip, the original flip's chance was still 1/2.<p>The possible misunderstanding comes from the fact that you will awake TWICE to Tails, which is more than to Heads! But the thing is this is irrelevant to the question, because you don't know whether this is the first or the second time, and you only need to wonder whether the original flip (recall it is probability 1/2) was heads or not. Potentially you could think that you'd be "wrong" more often, but we are only looking at one specific instance of you waking up in isolation, for which you have no additional information.<p>For example, consider waking up 1000 times if the flip is tails. Upon waking up, do you think the probability heads becomes 1/1001? I think regardless of waking up the first time or the 1000th time, you have no more information so it might as well have been the first time, and hence the probability is 1/2.
It seems to me that the paradox/confusion here comes from asking someone to consider a distribution over a bound variable - i.e. you are asked to examine the odds of X happening in a situation where there are already side effects of whether or not X happened.<p>In this sense, it reminds me of the puzzle where a man gives you a choice between two envelopes, one of which is specified to contain twice as much money as the other, with the paradox centering on a bystander's argument that you should then switch envelopes, since the other one must have either half or twice your value, giving you a 1.25x higher expected value for switching.<p>As I understand it, in both cases the "traditional" solution to the problem is to recognize that probability doesn't work that way, and you can't consider distributions over bound variables, but the more interesting solution is to rephrase things in Bayesian terms, in which case the analysis is reasonably straightforward.<p>I'm a dabbler though; experts please tell me if I'm spouting gibberish.
You're flipping a fair coin again and again. Every time you flip heads, you add 1 black ball to a (initially empty) bag. Every time you flip tails, you add 2 red balls to the same bag bag.<p>After a large number of flips, you pick a ball randomly from the bag.<p>What are the odds that the ball was added when a heads was flipped?<p>What are the odds that the ball is black?
I'm a layman in probability but isn't it the same thing as the Inspection paradox?
<a href="http://allendowney.blogspot.be/2015/08/the-inspection-paradox-is-everywhere.html" rel="nofollow">http://allendowney.blogspot.be/2015/08/the-inspection-parado...</a>
I remember reading rec.puzzles occasionally in the day when the Sleeping Beauty question was first posed. Coming back after not reading the group was for a couple of weeks was incredibly confused because it was full of minor variants of this single question constructed by people to support their position. I'd not only missed the original question and was thus lacking the context completely, but also had missed the megathread that followed and didn't understand the political undercurrents which would have been obvious if I'd only known who was halfer and who a thirder. It was just crazy.<p>Here's a great account of how it unfolded: <a href="http://www.maproom.co.uk/sb.html" rel="nofollow">http://www.maproom.co.uk/sb.html</a>
This reminds me of the Monty Hall problem, but in reverse - you were playing the Monty Hall game and just won the car! You forgot whether or not you switched doors during the second step - what is the probability you switched doors?
I'm not sure this is a paradox. There are 2 expected outcomes. All the added outcomes rely on the experimenter to disrespect the set rules.<p>This is a conditional probability with a hard to predict condition (ie. human factor) portrayed as a non-conditional probability. This looks more like a bad representation of a problem rather than a paradox.<p>This reminds me of my youth. Regardless of the project I was trying to accomplish, as soon as my little brother got involved, all bets where off. He was a hard to predict little bugger.
Even if you get awakened 99 times with Tails, you have equal probability going down the Heads (1 awakening) or Tails (1/99 awakenings) paths.<p>So if you awaken with no knowledge of other awakenings, you are equally likely to be on either path, with the coin being Heads or Tails.<p>You are NOT equally likely to guess that your awakening was due to Heads or Tails however, and that's the paradox.
I'm not sure I understand this given the way the question is posed.<p>We're doing a single experiment, and I am put to sleep without remembering either once or twice, and then awaken (potentially a third time) after that?<p>Or could this keep going indefinitely until heads comes up?<p>And am I guessing each time I'm woken, or only after the final time?
How do the researchers get you back to sleep for the second (tails) awakening?<p>You're woken once or twice and given the potion. Or you're woken and given the potion each time. This riddle wants it both ways, and that is part of the problem with it.