c4countdown

On Sunday you are given a drug that sends you to sleep. While you are asleep a fair coin is tossed. If it lands on heads, you are woken up on Monday and that's it. Job done. But if it lands on tails, you are woken up on Monday and then given another dose of the drug, which also causes you to forget about Monday's waking, and then you are woken up on Tuesday.

You wake up (you don't know if it's the first and only, first of two or second of two). What is the probability that the coin landed on heads?

Half. Now tell me what stupid counter-intuitive probabilistic nonsense is changing this.

1/3.

It's got to be half.

Suppose you were given a lethal drug, and only given the antidote if a coin came up heads. If you woke, you'd know it was heads. But that doesn't mean it would come up heads every time.

Presumably if it lands on heads, you'll still wake up on Tuesday, and whether it lands heads or tails, you'll still wake up on Wednesday, Thursday, etc. So 1/2.

Or are you only considering the times you wake up, thinking it's the first time you've woken up since Sunday? In which case I would go with 1/3.

David Williams wrote:It's got to be half.

Suppose you were given a lethal drug, and only given the antidote if a coin came up heads. If you woke, you'd know it was heads. But that doesn't mean it would come up heads every time.

It would come up heads every time given that you woke up. So from your point of view (which is what the question asks for), it would be heads with probability 1. Same idea with the original question.

Charlie Reams wrote:

David Williams wrote:It's got to be half.

Suppose you were given a lethal drug, and only given the antidote if a coin came up heads. If you woke, you'd know it was heads. But that doesn't mean it would come up heads every time.

It would come up heads every time given that you woke up. So from your point of view (which is what the question asks for), it would be heads with probability 1. Same idea with the original question.

I read the initial question, fired off an answer, and went to bed. As I was doing my teeth it occurred to me that if I'd proved anything, it wasn't what I thought I'd proved.

The coin is tossed once. It is a fair coin. Therefore the probability is half. QED?

Ian Volante wrote:The coin is tossed once. It is a fair coin. Therefore the probability is half. QED?

Nope. The later information can change the probabilities. For comparison, "I toss a fair coin and (truthfully) tell you it came up heads. What is the probability that it came up heads?" As common people would say, "not 'alf".

Charlie Reams wrote:

Ian Volante wrote:The coin is tossed once. It is a fair coin. Therefore the probability is half. QED?

Nope. The later information can change the probabilities. For comparison, "I toss a fair coin and (truthfully) tell you it came up heads. What is the probability that it came up heads?" As common people would say, "not 'alf".

Point taken, although this statement gives extra information, whereas the problem above doesn't necessarily do that.

Having looked up the solution, I think the problem could have been phrased better!

Ian Volante wrote:Point taken, although this statement gives extra information, whereas the problem above doesn't necessarily do that.

It tells you that you woke up. That's extra information.

Ian Volante wrote:Having looked up the solution, I think the problem could have been phrased better!

How?

In case anyone is still struggling - and it took me a while to be convinced.

If it's heads you'll wake up once, if it's tails you'll wake up twice. If you can persuade someone to enter into a wager where you give them £1 every time you wake if it was heads, and they give you £1 if it was tails, they'll pay you £2 for every £1 you pay them.

I'm with Ian on this one. No matter how much extra information you give me, the probability is always going to be 1/2. It's a fair coin. End of.

Charlie Reams wrote:

Ian Volante wrote:Point taken, although this statement gives extra information, whereas the problem above doesn't necessarily do that.

It tells you that you woke up. That's extra information.

Ian Volante wrote:Having looked up the solution, I think the problem could have been phrased better!

How?

It's not clear that you need to consider the problem from two different points of view, each of which give a different answer. I can see now that there's an implication that it should be looked at as if I'd just woken up, but taken in isolation, the actual question has no wording at all that makes this clear.

Kirk Bevins wrote:I'm with Ian on this one. No matter how much extra information you give me, the probability is always going to be 1/2. It's a fair coin. End of.

So you're saying that extra information can't affect the original probabilities? Let's look at another example.

I toss a fair coin. What's the probability that it came up heads? Half, obviously. Now, my friend Terrance (who has excellent vision) says he saw it was tails just before it landed. What's the probability now? Clearly not half.

The coin is still fair. However, there's other information available to you, and that information does affect the probabilities. The same process is at work in Gevin's problem.

Ian Volante wrote:I can see now that there's an implication that it should be looked at as if I'd just woken up, but taken in isolation, the actual question has no wording at all that makes this clear.

Except...

Gevin Chipper wrote:You wake up

Charlie Reams wrote:

Ian Volante wrote:I can see now that there's an implication that it should be looked at as if I'd just woken up, but taken in isolation, the actual question has no wording at all that makes this clear.

Except...

Gevin Chipper wrote:You wake up

That's hidden behind the full stop, hence not part of the question

Kirk Bevins wrote:I'm with Ian on this one. No matter how much extra information you give me, the probability is always going to be 1/2. It's a fair coin. End of.

So you repeat the experiment 100 times. 50 times the coin is heads, and each time you wake up once. 50 times the coin is tails, and each time you wake up twice. So you wake up 150 times altogether. One third of those times the coin was heads.

David Williams wrote:

Kirk Bevins wrote:I'm with Ian on this one. No matter how much extra information you give me, the probability is always going to be 1/2. It's a fair coin. End of.

So you repeat the experiment 100 times. 50 times the coin is heads, and each time you wake up once. 50 times the coin is tails, and each time you wake up twice. So you wake up 150 times altogether. One third of those times the coin was heads.

Good explanation, yes.

Charlie Reams wrote:

David Williams wrote:

Kirk Bevins wrote:I'm with Ian on this one. No matter how much extra information you give me, the probability is always going to be 1/2. It's a fair coin. End of.

So you repeat the experiment 100 times. 50 times the coin is heads, and each time you wake up once. 50 times the coin is tails, and each time you wake up twice. So you wake up 150 times altogether. One third of those times the coin was heads.

Good explanation, yes.

The thing is I've never been entirely happy with either answer to this problem. Yes, the 1/3 answer makes sense from this perspective and I'd considered what would happen if you bet on it. But one could also argue that it's still 1/2 but that you're going to put twice as much money down in the wrong cases.

My main problem though is this - what if after you go to sleep, the national lottery takes place and if the experimenter wins, he wakes you up 28 million times, otherwise it's just once. There's about 14 million combinations so it should be about the same - 1/3 chance he hasn't won the lottery and 2/3 chance he has. But wouldn't you be surprised anyway if he had won?

The key here is that if the coin lands on tails, you wake up twice. If heads, you wake up only once. So when you wake up, 2 times out of 3 the coin will have landed on tails.

I still can't get my head around this though (the second problem). Probability is mind-boggling.

Edit: After a re-read of some explanations, I get the second problem.

Charlie Reams wrote: So you're saying that extra information can't affect the original probabilities? Let's look at another example.

I toss a fair coin. What's the probability that it came up heads? Half, obviously. Now, my friend Terrance (who has excellent vision) says he saw it was tails just before it landed. What's the probability now? Clearly not half.

The coin is still fair. However, there's other information available to you, and that information does affect the probabilities. The same process is at work in Gevin's problem.

This is surely nonsense. That's like saying it comes up heads 50% of the time but Terrance is a bit blind and says tails all the time. It doesn't change the probability of the coin.

I bet if you did Gevin's experiment in real life, heads would still come up 50% of the time. If it only comes up 1/3 of the time, I'll buy you a pint.

David Williams wrote: So you repeat the experiment 100 times. 50 times the coin is heads, and each time you wake up once. 50 times the coin is tails, and each time you wake up twice. So you wake up 150 times altogether. One third of those times the coin was heads.

Nice explanation, but 50 times it was heads, 50 times tails, so it's 1/2. Probability is really really stupid.

Gavin Chipper wrote: My main problem though is this - what if after you go to sleep, the national lottery takes place and if the experimenter wins, he wakes you up 28 million times, otherwise it's just once. There's about 14 million combinations so it should be about the same - 1/3 chance he hasn't won the lottery and 2/3 chance he has. But wouldn't you be surprised anyway if he had won?

You might be surprised, but that's a purely intuitive reaction based on the perception of winning the lottery as an improbable event. Humans are notoriously bad at intuiting probability.

Kirk Bevins wrote:Nice explanation, but 50 times it was heads, 50 times tails, so it's 1/2. Probability is really really stupid.

I think I see where you're going wrong here. The question isn't "how often is the coin heads", which is what you're answering. It's "Given that you've just been woken up, how often was the coin heads?"

To possibly repeat an earlier point, imagine that you aren't woken up at all when it's tails. Given that you've just been woken up, how often was the coin heads? 100% of course. This doesn't contradict the coin being fair.

I hate paradoxes.

Charlie Reams wrote:

Gavin Chipper wrote: My main problem though is this - what if after you go to sleep, the national lottery takes place and if the experimenter wins, he wakes you up 28 million times, otherwise it's just once. There's about 14 million combinations so it should be about the same - 1/3 chance he hasn't won the lottery and 2/3 chance he has. But wouldn't you be surprised anyway if he had won?

You might be surprised, but that's a purely intuitive reaction based on the perception of winning the lottery as an improbable event. Humans are notoriously bad at intuiting probability.

OK, but before we leave intuitive reactions, imagine that if it's heads they wake you up once and if it's tails they wake you up a million times. Would you be freaked out to find that it was heads?

On the other hand, if someone argues it's 1/2, what happens if the coin isn't tossed until the first time they wake you up? The experiment will reach that point in the same way anyway, so they could do that. So you wake up. If it's the first time you wake up, then the coin toss has a 50/50 chance of going each way. But there's also a chance that it will be the second time you wake up, in which case it's definitely tails. Which means that tails is more likely!

Is this vaguely similar to the Monty Hall problem?

David O'Donnell wrote:Is this vaguely similar to the Monty Hall problem?

Not really. I don't think so anyway. I think this is more "philosophical" and I'm not sure that it's simply a case of proving that one answer is correct here.

When you toss a fair coin it comes up 50:50 - no question, no argument. This is the definition of "fair". What you afterwards do with the decision is entirely irrelevant. If Gavin seriously claims otherwise I'm guessing that he worded the question a trifle more loosely than he intended.

David O'Donnell wrote:Is this vaguely similar to the Monty Hall problem?

I think it is. That's another case where the odds are entirely clear yet people argue about it without end. The solution is for somebody to write a simulation of the game. Let the computer run through it a few thousand times and the odds will become clear to everybody.

Rosemary Roberts wrote:When you toss a fair coin it comes up 50:50 - no question, no argument. This is the definition of "fair". What you afterwards do with the decision is entirely irrelevant. If Gavin seriously claims otherwise I'm guessing that he worded the question a trifle more loosely than he intended.

You're misreading it in the same way that Kirk is misreading it, which I already addressed, so I won't repeat myself.

Rosemary Roberts wrote:

David O'Donnell wrote:Is this vaguely similar to the Monty Hall problem?

I think it is. That's another case where the odds are entirely clear yet people argue about it without end. The solution is for somebody to write a simulation of the game. Let the computer run through it a few thousand times and the odds will become clear to everybody.

Just to humour you, here's a simulation I wrote. The princess is awoken 904 times and the coin was heads 296 of those, which is 32.7%. I ran another for 1,000,000 iterations and that was 33.3% (can't be arsed uploading that one). I think you get the idea.

Charlie Reams wrote: Just to humour you, here's a simulation I wrote. The princess is awoken 904 times and the coin was heads 296 of those, which is 32.7%. I ran another for 1,000,000 iterations and that was 33.3% (can't be arsed uploading that one). I think you get the idea.

I'm impressed, and of course surprised. However, a fair coin is fair and so my argument isn't wrong. What evidently is wrong is that I didn't study the question closely enough: not all coin tosses are included.

I said a simulation would clear it up. Thanks, Charlie.

Rosemary Roberts wrote:

Charlie Reams wrote: Just to humour you, here's a simulation I wrote. The princess is awoken 904 times and the coin was heads 296 of those, which is 32.7%. I ran another for 1,000,000 iterations and that was 33.3% (can't be arsed uploading that one). I think you get the idea.

I'm impressed, and of course surprised. However, a fair coin is fair and so my argument isn't wrong. What evidently is wrong is that I didn't study the question closely enough: not all coin tosses are included.

I said a simulation would clear it up. Thanks, Charlie.

No problem. FWIW all tosses are included, it's just that some of them (the tails) are effectively counted twice.

Tbh I think this thread is in incredibly poor taste - it is clearly a metaphor for the 'rohypnol prostitution' that is pervading Broken Britain. To try and turn such a serious issue into a 'puzzle' where people argue over the minutiaes of a sentence is despicable and Charlie should be banned and I have made a formal complaint to the moderators.

Gavin Chipper wrote:

David O'Donnell wrote:Is this vaguely similar to the Monty Hall problem?

Not really. I don't think so anyway. I think this is more "philosophical" and I'm not sure that it's simply a case of proving that one answer is correct here.

Have to state that after a restless sleep I am starting to agree with you. In fact on further research it seems the problem is well known to philosophy. I shall have to look at it more closely though I find the reasoning hard to grasp since I study 'continental' philosophy rather than 'analytic' philosophy.

I don't think the answer is wrong, so much as the question is phrased wrongly.

At any given time, the odds of the coin having come down heads last time it was tossed is 1/2. And if this experiment takes place only once, then similarly, the answer is 1/2 - because what happens after the initial coin toss has no bearing on the original result.

If the question had been, "in a series of repeat experiments the coin is tossed many times and one of the awakenings of the princess is chosen at random, then what is the probability the coin was heads", then you'd have 1/3.

[Edit to add] or in a single experiment, the question should have been "what is the probability that this awakening has happened after the coin toss was tails". It's not the same thing.

In the case where they wake you a million times if it's tails, then if we should be surprised that it's heads, we should we also be surprised to be told that it is the end of the experiment at any point (that also being very unlikely). And given that the experiment will end, you're going to end up pretty surprised at some point - and when it does happen, heads is 50/50.

Considering the million case again - they wake you up, you decide that it's virtually certain to be tails, but it turns out to be heads and the experiment ends there and then. And you think "how weird - I was expecting tails." But then it turns out that if it had landed on tails they might not have proceeded with the rest of the experiment anyway. There is a sealed box that has remained sealed for thousands of years. All anyone knows is that it has either a red or a blue stone in it. If it had landed on tails, they would have opened the box. If the stone was blue, the would have only woken you up once and ended the experiment. But a red stone would have meant the experiment being completed properly. So you think "Oh, so that's why it ended up as heads despite the massive odds against. There must be a blue stone in the box." Is this valid reasoning? No-one has a clue what's in the box and it hasn't affected anything.

Further thoughts:

It's a fair coin, so on Sunday it's a 50/50 chance whether it's heads or tails.

There are up to two awakenings.

On Monday, the probability of being awoken is 1.
1a. The probability of being awoken after a head is 1.0 * 0.5 = 0.5. And given that it was a head and there's no second awakening, then the probability of being asked whether it was a head or tail is 0.5 * 1.0 = 0.5.
1b. The probability of being awoken after a tail is similarly 0.5. And with there being a second awakening due, the probability of being asked whether it was a head or a tail is 0.5 * x = 0.5x, where x is an unknown between 0 and 1.

2. On Tuesday, the probability of being awoken given that you're still asleep is 1.0 x 0.5 = 0.5, and the probability that you're asked whether it was a head or a tail given that you've just been woken is 0.5*(1-x) = 0.5 - 0.5x.

So add the three together, and you get from 1a, a 0.5 chance that the coin toss was a head. And from 1b + 2, a (0.5x + .05 - 0.5x) = 0.5 chance the coin was a tail. Which is what you would expect from a fair coin. The paradox is in the question setting. You can only get a 1/3 answer by listing all possible awakenings and picking one at random - and that isn't what the question asked.

David Roe wrote:And given that it was a head and there's no second awakening, then the probability of being asked whether it was a head or tail is 0.5 * 1.0 = 0.5.

You have misread the question. Given that it was a head and there's no second awakening, the probability of being "asked" (a term you have introduced which was not in the question) whether it was a head or tail is 1, not 0.5. You always get "asked" in the sense that each time you wake up is (from your perspective) a new trial. The question says

Gevin wrote:You wake up. What is the probability that the coin landed on heads?

Gavin Chipper wrote:In the case where they wake you a million times if it's tails, then if we should be surprised that it's heads, we should we also be surprised to be told that it is the end of the experiment at any point (that also being very unlikely). And given that the experiment will end, you're going to end up pretty surprised at some point - and when it does happen, heads is 50/50.

Considering the million case again - they wake you up, you decide that it's virtually certain to be tails, but it turns out to be heads and the experiment ends there and then. And you think "how weird - I was expecting tails." But then it turns out that if it had landed on tails they might not have proceeded with the rest of the experiment anyway. There is a sealed box that has remained sealed for thousands of years. All anyone knows is that it has either a red or a blue stone in it. If it had landed on tails, they would have opened the box. If the stone was blue, the would have only woken you up once and ended the experiment. But a red stone would have meant the experiment being completed properly. So you think "Oh, so that's why it ended up as heads despite the massive odds against. There must be a blue stone in the box." Is this valid reasoning? No-one has a clue what's in the box and it hasn't affected anything.

Bit tl;dr for me but at first glance this sounds a lot like the Unexpected Hanging Paradox.

Charlie Reams wrote:FWIW all tosses are included, it's just that some of them (the tails) are effectively counted twice.

Oh bugger. Not my day. Not my week.

Nick Bostrom has some interesting remarks on this problem here.

Charlie Reams wrote:You have misread the question. Given that it was a head and there's no second awakening, the probability of being "asked" (a term you have introduced which was not in the question) whether it was a head or tail is 1, not 0.5. You always get "asked" in the sense that each time you wake up is (from your perspective) a new trial. The question says

Gevin wrote:You wake up. What is the probability that the coin landed on heads?

Charlie, please don't equate "disagrees with you" to "a bit thick". I haven't misread the question, I've read it several times.

As you more-or-less accurately put it, the question is "what is the probability that the coin landed on heads?" To which, in my opinion, the answer is 0.5. It was 0.5 at the time, it remains 0.5 no matter how many times this person has been drugged. If the question had been "what is the probability that this awakening took place when the last coin toss was a head?", then the answer would have been 1/3.

All this question is doing is asking the question "Heads or Tails" after the toss of the coin. If the coin comes down heads, I ask the question once; if it comes down tails, I ask twice. The probability of the correct answer being heads is 1/3. But the probability of it coming down heads is 1/2. IMO.

David Roe wrote:All this question is doing is asking the question "Heads or Tails" after the toss of the coin. If the coin comes down heads, I ask the question once; if it comes down tails, I ask twice. The probability of the correct answer being heads is 1/3. But the probability of it coming down heads is 1/2. IMO.

I think Charlie's simulation is pretty conclusive, but before that I had to think of it in terms of possible outcomes, in that it could be Monday/heads, Monday/tails or Tuesday/tails (not Tuesday/heads as the experiment would have ended on Monday). So if the coin is fair, then any of the three outcomes is as likely from the princess's perspective, so it's 1/3 for heads.

Hmm, having read that back I'm not sure that's quite right but it makes sense to me.

JimBentley wrote:

David Roe wrote:All this question is doing is asking the question "Heads or Tails" after the toss of the coin. If the coin comes down heads, I ask the question once; if it comes down tails, I ask twice. The probability of the correct answer being heads is 1/3. But the probability of it coming down heads is 1/2. IMO.

I think Charlie's simulation is pretty conclusive, but before that I had to think of it in terms of possible outcomes, in that it could be Monday/heads, Monday/tails or Tuesday/tails (not Tuesday/heads as the experiment would have ended on Monday). So if the coin is fair, then any of the three outcomes is as likely from the princess's perspective, so it's 1/3 for heads.

Hmm, having read that back I'm not sure that's quite right but it makes sense to me.

I think the arguments for 1/2 are all to do with the phrasing of the question.

JimBentley wrote:

David Roe wrote:All this question is doing is asking the question "Heads or Tails" after the toss of the coin. If the coin comes down heads, I ask the question once; if it comes down tails, I ask twice. The probability of the correct answer being heads is 1/3. But the probability of it coming down heads is 1/2. IMO.

I think Charlie's simulation is pretty conclusive, but before that I had to think of it in terms of possible outcomes, in that it could be Monday/heads, Monday/tails or Tuesday/tails (not Tuesday/heads as the experiment would have ended on Monday). So if the coin is fair, then any of the three outcomes is as likely from the princess's perspective, so it's 1/3 for heads.

Hmm, having read that back I'm not sure that's quite right but it makes sense to me.

How confident are you that the stone in the box is red?

JimBentley wrote:

David Roe wrote:All this question is doing is asking the question "Heads or Tails" after the toss of the coin. If the coin comes down heads, I ask the question once; if it comes down tails, I ask twice. The probability of the correct answer being heads is 1/3. But the probability of it coming down heads is 1/2. IMO.

I think Charlie's simulation is pretty conclusive, but before that I had to think of it in terms of possible outcomes, in that it could be Monday/heads, Monday/tails or Tuesday/tails (not Tuesday/heads as the experiment would have ended on Monday). So if the coin is fair, then any of the three outcomes is as likely from the princess's perspective, so it's 1/3 for heads.

Hmm, having read that back I'm not sure that's quite right but it makes sense to me.

It does seem to be extremely complicated and I don't really want to join in until I have understood it better but since eminent philosophers of logic and probability seem to disagree I doubt that will be any time soon.

I would just suggest that Charlie's simulation may be guilty of confirmation bias as it will, over time, even out at 1/3 for heads: there are three outcomes after all. The baffling step in the logic is to ascribe an equal probability for each awakening upon which the 1/3 argument rests. Also, I think Gevin gave the example earlier, if we were to wake Beauty up on one million subsequent days would this mean that the probability that the coin lands on heads be 1/1,000,002.

I think because the problem has different Beautys in various spatio-temporal locations involving a conflation of various partially indexable locations allows paradoxes to emerge. The 1/2 argument also leads to paradoxical conclusions.

Bostrom's argument is interesting because he adopts a hybrid approach between the two philosophical positions that seems to be consistent in advocating an approach for Beauty to correctly assess her odds when faced with betting payouts from a bookie. I have only read through his paper once though and suspect it requires more careful analysis.

David Roe wrote:

Charlie Reams wrote:You have misread the question. Given that it was a head and there's no second awakening, the probability of being "asked" (a term you have introduced which was not in the question) whether it was a head or tail is 1, not 0.5. You always get "asked" in the sense that each time you wake up is (from your perspective) a new trial. The question says

Gevin wrote:You wake up. What is the probability that the coin landed on heads?

Charlie, please don't equate "disagrees with you" to "a bit thick".

I didn't say you were a bit thick, but if you choose to be insulted by criticism of your argument then that's up to you.

David Roe wrote: I haven't misread the question, I've read it several times.

Non sequitur.

David Roe wrote:As you more-or-less accurately put it, the question is "what is the probability that the coin landed on heads?"

No it isn't.

David Roe wrote:All this question is doing is asking the question "Heads or Tails" after the toss of the coin. If the coin comes down heads, I ask the question once; if it comes down tails, I ask twice. The probability of the correct answer being heads is 1/3. But the probability of it coming down heads is 1/2. IMO.

If that were really the case then Beauty should always be prepared to bet at odds of 5/4 that the coin is heads. Would you take that bet?

David O'Donnell wrote:I would just suggest that Charlie's simulation may be guilty of confirmation bias as it will, over time, even out at 1/3 for heads: there are three outcomes after all.

Did you read the file I linked to? I didn't program it to just pick one of the three outcomes at random (and if I had done, that would be a better example of pure idiocy than confirmation bias). It's just a repeated simulation of the experiment as described, counting up the frequency that Beauty sees heads. The only constant in the program is, in fact, 0.5 (from the fair coin).

David O'Donnell wrote:The baffling step in the logic is to ascribe an equal probability for each awakening upon which the 1/3 argument rests. Also, I think Gevin gave the example earlier, if we were to wake Beauty up on one million subsequent days would this mean that the probability that the coin lands on heads be 1/1,000,002.

That document looked interesting but it had a lot of unfamiliar terminology so I got lost at some point.

I got fairly lost too as I stated before the analytical philosophers are like a world apart from us but I am trying to understand them better because Lyotard has elements of both traditions. I get the general gist of the argument but some of the jumps, he makes, leaves me scratching my head.

I think it's sufficient to note that there is significant philosophical dispute about this problem but that this may be on a philosophical level: the mathematical problem may be different: I really don't know!

At first I was fairly convinced by your argument but Gevin may have a point that there is a philosophical dispute here too.

I wonder if part of the difficulty here is that there is more than one party involved, and that Beauty does not know the outcome of the coin toss.

I use a somewhat unusual circular train line. There are only three stations, trains run in both directions, on the hour in one direction and on the half hour the other way. I get to my station at random times. The stations are twenty minutes apart, so it makes sense to get the train going the wrong way, which will take forty minutes to get to your destination, rather than wait half an hour for one going the right way. On the train I doze and am woken when the train stops.

Clearly the chances of going one way or the other are equal. But only one-third of the times that I wake will I be on the 'right' train. Would anyone disagree with that?

Maybe it comes down to what people understand by "probability". The very specific meaning used by mathematicians may not be shared by everyone.

If there are two hypotheses - one that Dave rides his bike for several hours each day and one that he rides it once a year, and you pick a random point in time and Dave happens to be riding a bike, then the hypothesis that he rides his bike every day looks more plausible. But if you specifically pick a point in time when he are riding a bike then it doesn't say very much. Similarly you know you're going to be waking up - there's nothing else that could have happened - so is it really 2/3 that it's tails? What new information have you learnt? When it gets to the final time you wake up, they might tell you this before they tell you whether it's heads or tails. At this point it's back to 50/50. What information have you now gained?

Also right at the bottom of this:

Here is a betting scenario that favours the halfers:

The bets are set up so that Sleeping Beauty gets $36 if she predicts the throw correctly at least once.

If she has deterministic thought processes, and must therefore answer the same in both cases, then the payout will be $18 regardless of which choice she makes (thirders take note).

However, if she can make truly random guesses at each waking (flip of a coin, perhaps!), then she will get the best payout by using the probability of 1/2, which gives her a payout of $22.50 (5/8 of $36). Using the probability of 1/3 would give her a payout of $22.00 (11/18 of 36).

So the "halfers" are just as justified as mathematically justified as the "thirders".

I haven't checked the maths or anything but this could be interesting.

David Williams wrote:Maybe it comes down to what people understand by "probability". The very specific meaning used by mathematicians may not be shared by everyone.

Thinking about this again, how many other mathematical situations are there where the probability of something is 1/2 but is guaranteed to change to 1/3 (or indeed any other definite amount) and then back to 1/2 again (if you're told when it's the final time you wake up before the result of the coin toss is revealed)?

Actually, I've just read that Dmitry Goretsky is a probability genius, so if you're reading this, what's your take on it?

I've still not got round to attempting to read the article David linked to.

Reading the original question, there is only one coin toss on the Sunday. It doesn't actually say there's another coin toss on the Monday to determine if you will be woken on the Tuesday. Only one coin toss means it's 50:50. Although according to Qi even fair coin tosses aren't quite 50:50 because whether heads or tails is facing up or down will have an affect on the outcome.

Mark James wrote:Reading the original question, there is only one coin toss on the Sunday. It doesn't actually say there's another coin toss on the Monday to determine if you will be woken on the Tuesday.

There isn't another coin toss.

Only one coin toss means it's 50:50.

Nope.

Mark James wrote:Reading the original question, there is only one coin toss on the Sunday. It doesn't actually say there's another coin toss on the Monday to determine if you will be woken on the Tuesday. Only one coin toss means it's 50:50. Although according to Qi even fair coin tosses aren't quite 50:50 because whether heads or tails is facing up or down will have an affect on the outcome.

Forget QI. In a fair coin toss, you don't know which of heads or tails is on top when you toss it.

As for the question, the probability of the coin landing on heads on Sunday was and is 1/2. The probability of an awakening happening after a head is 1/3. The rest of it is just arguing about the wording of the question.

There are three points at which you can be woken up:

- On Monday after a heads toss
- On Monday after a tails toss, or
- On Tuesday after a tails toss.

In two of these three cases the coin landed on tails.

Given that you have just woken up, it is twice as likely that the coin landed on tails. Probability (Heads) = 1/3.