The Sleeping Beauty Problem: What is the Scientific Definition of Credence?

[Moes]

TL;DR Summary

What is the argument of a third?

Many people debate the sleeping beauty problem, and seemingly there are many who take the thirder view. I don’t see how anyone can argue that the answer is anything but 1/2. Before she goes to sleep the probability of heads or tails is 50/50. The only argument I can think of, to say that anything changes when she is put to sleep and wakes up, is that she could think to herself what were the chances of her being woken up that day? Well if the coin landed heads it was only 50/50, so it would be a slight surprise that she was woken up that day, so shouldn’t she assume it landed tails in which case she was for sure going to be woken up? Well I don’t see this as an argument since the only time she could think to herself about the chances of her being woken up is the one time she is woken up, so it’s no surprise. Even if she was to be woken up just once in the next million days there is no surprise on the day she wakes up. She always knew there was a day she was going to be woken up. Therefore, she doesn’t get any new information when she wakes up, so her credence that the coin landed heads should still be 1/2.
As for the argument against the halfer view that if she is told it is Monday when she wakes up the chances of heads or tails should be 50/50, I think this is obviously wrong. If the coin landed tails it’s slightly surprising that it’s Monday, but if it was heads it was obviously Monday. So the odds should be that the coin probably landed heads. It’s easy to see this through the extreme case where she is woken up for the next million days if the coin lands tails. If she is told when she is woken up that it happens to be the one Monday that she would have been woken up whether the coin landed heads or tails, she should obviously assume the coin landed heads.
So what is the argument for the thirder position?

 

[DaveC426913]

For those of us that need context:
https://en.wikipedia.org/wiki/Sleeping_Beauty_problem

 

[hutchphd]

Can we call Monty Hall for an explanation?

.

 

[PeroK]

So what is the argument for the thirder position?

The argument for 50-50 is the one that you have given: it must be 50-50 and no rational person would disagree.

The argument for a third is to do the calculations and find that the answer is one third. There are similar problems where the answer is one third by the same calculations. The 50-50 position, however, is that you are not allowed to do calculations or use mathematics in this specific problem.

This is based on the dubious logic that the sleeping beauty is a princess, not a mathematician, and not capable of mathemetical calculations.

We had a long thread on this a few years ago that turned rather nasty and almost led to my leaving PF.

 

[PeroK]

@Moes let me give you a different problem and see what you think of this.

You toss a fair coin and ask what is the probability that it is heads or tails. The answer is 50-50.

Then you look at the coin and see that it is Heads, say. You now ask the question again: what is the probability that it was heads or tails. There are now two answers depending how you interpret the question:

1) The answer is still 50-50, since looking at the coin does not change the inherent probability of a fair coin, landing heads or tails. The probability was, is and will always remain 50-50.

2) The answer is now certain that it was heads: 100%

The sleeping beauty princess problem is similar, although definitely more subtle. And, of course, answer 2) is one third and not certain.

 

[Dale]

I don’t see how anyone can argue that the answer is anything but 1/2. Before she goes to sleep the probability of heads or tails is 50/50.

Suppose that the experiment is changed slightly. They will flip the coin and leave it untouched where it lands so that she can inspect it. She is woken and shown the coin and asked her credence that it is heads.

As in your case, before she goes to sleep the probability is 50/50. Do you still think her credence should be 1/2 after she sees the coin?

 

[Filip Larsen]

Interesting problem, but I struggle to follow the argument behind the Bayes approach here.

What information does the sleeping beauty gain when she is awoken? As I see it, nothing, since there is no way she can discern what day it is. And if she is given no new information (in addition to the knowledge that a coin was indeed flipped) how can the calculation change? Why does it change her belief about heads or tails that she is (potentially) asked multiple times (with no memory of being asked in between)?

Note that in the Monty Hall problem the quiz contestant *is* given new information, which he then can use to arrive at a "better" estimate of probabilities given his updated knowledge.

 

[Dale]

What information does the sleeping beauty gain when she is awoken? As I see it, nothing, since there is no way she can discern what day it is.

Information does not require that you be able to definitely determine a hypothesis. It only requires that the information be more likely under one hypothesis than under the other. In this case the fact that she is awoken is twice as likely under one hypothesis than under the other. So the fact that she is awoken is indeed information.

 

[PeroK]

Interesting problem, but I struggle to follow the argument behind the Bayes approach here.

What information does the sleeping beauty gain when she is awoken? As I see it, nothing, since there is no way she can discern what day it is. And if she is given no new information (in addition to the knowledge that a coin was indeed flipped) how can the calculation change?

Another way to look at it is that her answer does not change! There are two different questions here, both of which may be asked at the beginning of the experiment:

1) What is your credence that its is heads?

2) When you are woken, what will be your credence that it is heads.

I claim that even initially, before she goes to sleep, the answer to question 2) is $1/3$. And, indeed, nothing changes. The answer to question 2) is always $1/3$.

 

[Filip Larsen]

So the fact that she is awoken is indeed information.

I do not see how she can gain any information by being awoken. She knows upfront she will be awoken at least once before the experiment ends so when she is awoken (without memory) she gains no new knowledge. She do know, however, that if today is Monday she will with probability 1 also be awoken Tuesday.

2) When you are woken, what will be your credence that it is heads.

I claim that even initially, before she goes to sleep, the answer to question 2) is 1/3. And, indeed, nothing changes. The answer to question 2) is always .

So, if the experiment is changed to last, say, 99 days instead of 2, then by that argument whenever she is awoken she should believe the coin is heads with probability 99%? That simply doesn't sound like a "proper" model of her belief given the only set of events is the string of exactly either 1 day awake or 99 days awake (there is no other sequence of events). Why does being asked 99 times about something make the outcome near certain?

What if we change the experiment so that when the coin is on heads she is only awaken each day with some independent probability p and look what happens in the limit as p goes to 1?

 

[PeroK]

So, if the experiment is changed to last, say, 99 days instead of 2, then by that argument whenever she is awoken she should believe the coin is heads with probability 99%? That simply doesn't sound like a "proper" model of her belief given the only set of events is the string of exactly either 1 day awake or 99 days awake (there is no other sequence of events). Why does being asked 99 times about something make the outcome near certain?

Let's forget the sleeping potion for a moment. You tell someone that if the coin is tails you will come and ask them 99 times whether it was heads or tails. If it's heads you will ask them only once.

It should be clear that the first time you ask them it's still 50-50, but on the subsequent 98 occasions it must be a tail. They'd be a fool to bet on heads after the first time of asking! (Although, not everyone accepts that probability equates to being able to break even on what you believe!)

Now, you ask them to consider the average probability. Assuming you allow me to repeat this experiment many times (which not everyone does, it must be said).

It should then be clear that, overall, when the question is asked, the probability is 99% that it was a tail.

IF you don't accept that, then it's not the sleeping princess problem that is the issue: it's basic conditional probability theory.

The question here is whether the amnesia-inducing sleeping potion makes a difference. In other words, if she cannot remember the previous occasions, can she still calculate in the knowledge that they might have taken place? Obviously, she has to remember (or be informed of) the rules of the game, otherwise (from her perspective) she has no idea what's going on.

If we were playing this game for real money, what would you do? Would you be happy to answer heads and be confident you would break even?

I'm certainly confident that if we played this game with the 99-day rule and you were willing to bet $100 per time on heads (and I bet on tails), then I'd clean you out. I don't see any way you could break even betting heads every time.

 

[Dale]

I do not see how she can gain any information by being awoken.

For the hypothesis that the flip was tails, the amount of information in bits obtained by being woken on either Monday or Tuesday can be calculated as follows: $$ \log_2 \left(\frac{P(wake|tails)}{P(wake|heads)}\right) = \log_2 \left(\frac{1}{0.5}\right)= 1 \text{ bit}$$

 

[Dale]

If we were playing this game for real money, what would you do? Would you be happy to answer heads and be confident you would break even?

This is a good point and is the usual way that credence is described. In this case, I think that describing a specific wager is useful because it can clarify what specific credence value is requested.

The “thirders” and the “halfers” are generally thinking about different wagers. A “thirder” is thinking of a straightforward wager made every time she is awoken and interviewed. A “halfer” is thinking of some more complicated wager. I find that if everyone discusses a specific wager then everyone agrees on the odds.

 

[PeroK]

This is a good point and is the usual way that credence is described. In this case, I think that describing a specific wager is useful because it can clarify what specific credence value is requested.

The “thirders” and the “halfers” are generally thinking about different wagers. A “thirder” is thinking of a straightforward wager made every time she is awoken and interviewed. A “halfer” is thinking of some more complicated wager. I find that if everyone discusses a specific wager then everyone agrees on the odds.

In this case, it's if the bet is only the first time she is asked. But, then the whole rigmarole of amnesia-inducing sleeping potions is a waste of time. Why introduce all that complexity, if none of it is relevant? To me, that's just a cop-out by the halfers when they know they're busted!

 

[Filip Larsen]

The question here is whether the amnesia-inducing sleeping potion makes a difference.

If she are allowed memory about whether or not she has been asked before and knows that up front, then it should make a difference. With such memory, first time she is asked she knows it is first time so probability for heads should be 1/2 on that occasion. If she is asked a second time she now knows it must be head with probability 1.

If we compare the these two experiments (with and without memory), then I think the argument I would like to pose is that that because she arrives at probability 1/2 first time asked with memory she should also arrive at 1/2 "first time" she is asked without memory, although in the latter "first time" is every time she is asked.

 

[Dale]

Why introduce all that complexity, if none of it is relevant?

I agree. Some of the wagers that “halfers” propose are hard to justify only in terms of the described scenario.

Nevertheless, if someone describes the wager they have in mind then everyone can agree on the odds, even if they don’t agree that said wager is relevant to the Sleeping Beauty problem description.

 

[Filip Larsen]

For the hypothesis that the flip was tails, the amount of information in bits obtained by being woken on either Monday or Tuesday can be calculated as follows: $$ \log_2 \left(\frac{P(wake|tails)}{P(wake|heads)}\right) = \log_2 \left(\frac{1}{0.5}\right)= 1 \text{ bit}$$

Why would you set $P(wake|heads) = 0.5$? I think this is the gist of the difference for me, as I would claim it makes more sense to have $P(wake|heads) = 1$ since the probability she is awaken (without memory) is 1 no matter what the coin says.

 

[Dale]

Why would you set P(wake|heads)=0.5?

Because if it is heads and Monday she is woken and if it is heads and Tuesday then she is not woken. Therefore the probability that she is woken on Monday or Tuesday given that it is heads is 0.5.

 

[PeroK]

If we compare the these two experiments (with and without memory), then I think the argument I would like to pose is that that because she arrives at probability 1/2 first time asked with memory she should also arrive at 1/2 "first time" she is asked without memory, although in the latter "first time" is every time she is asked.

That's a valid calculation. If there are things you don't know, then you can lose money on what you think is a 50-50 bet. If she knows nothing about the game (or has forgotten), then she can only guess 50-50. But, dare I say it, that is trivial. There is no fundamental controversy over that point.

The interesting question - and the only reason to pose this problem - is where she does know the rules of the game (or is reminded of them) but cannot remember the events that have so far taken place. If you don't have that, then there is nothing of any interest.

What I argue is that if you put me in the role of sleeper, then I would not lose money (as long as I was told the rules of the game). Even if I couldn't remember what had happened, I would guess tails. In other words, I claim I could still get the probabilities "right" and not lose money. Whereas, if I don't know the rules of the game and am forced to bet, then I'll lose money on average.

Your analysis condemns me (the sleeper) to losing money by giving me a probability of 50-50. In fact, that would be really funny. You'd be telling me that the probability I should calculate is 50-50 (with your full knowledge) and I'd be arguing with you (even with only partial knowledge) that it's $1/3 - 2/3$.

You are then forced to tell me that I'm wrong to get the right answer!

 

[Filip Larsen]

Dang it, I just realize I have head and tails flipped, i.e. I thought she was asked about the probability for the coin landing on the outcome where she will be awoken multiple times But even with this correction, if I thought her best answer was 1/2 before it should still be 1-1/2 = 1/2 after. I think I need to reread the thread and try argue with proper coin "orientation" if needed.

 

[PeroK]

Dang it, I just realize I have head and tails flipped, i.e. I thought she was asked about the probability for the coin landing on the outcome where she will be awoken multiple times But even with this correction, if I thought her best answer was 1/2 before it should still be 1-1/2 = 1/2 after. I think I need to reread the thread and try argue with proper coin "orientation" if needed.

I don't think that matters. The summary of the thirder position is this. When the sleeper is woken:

1) Someone who knows what is happening can bet correctly 100% of the time. I.e. they know whether it was a head or tail.

2) Someone who knows the rules of the game but doesn't know what has happened so far can be correct two-thirds of the time by guessing tails every time.

3) Someone who knows neither the rules of the game nor what has happened so far can only be correct one-third of the time by guessing either heads or tails randomly.

Your argument, if I am allowed to put words in your mouth, is that in the case of 2) no one can be that clever and they must be reduced to guessing 50-50, as someone in case 3). I, for one, claim to be clever enough to be correct two-thirds of the time in case 2).

 

[Filip Larsen]

If we were playing this game for real money, what would you do?

I think I see your point with the gaming questions now. If we change the experiment so she either wins or loose 1$ if she can guess the coin face correctly each time she is awaken, then she will on average win 1$ by guessing on tails.

But is that really the same "rules" as simply asking "to the best of your knowledge right now, that is the probability the coin was heads"? By summing wins and losses you have added some kind of "memory" to the experiment. Or in other words, I can't see that just because tails is a better choice in the game version that this somehow should imply that P(head) = 1/3 (in either versions).

 

[PeroK]

But is that really the same "rules" as simply asking "to the best of your knowledge right now, that is the probability the coin was heads"? By summing wins and losses you have added some kind of "memory" to the experiment. Or in other words, I can't see that just because tails is a better choice in the game version that this somehow should imply that P(head) = 1/3 (in either versions).

If probabilities don't relate to betting (outcomes in terms of relative frequency), then what are they?

I might claim that if we roll a die, then the probability of rolling a six is $0.5$.

You can try to prove me wrong by rolling the die repeatedly and showing that it comes up six with a relative frequency of $1/6$ and not $1/2$.

But, I can play your argument back to you and say:

Just because "not six" is a better choice in the game version this does not imply that P(six) $\ne 1/2$.

 

[Dale]

I can't see that just because tails is a better choice in the game version that this somehow should imply that P(head) = 1/3 (in either versions).

That is the usual operational definition of credence.

https://en.m.wikipedia.org/wiki/Credence_(statistics)

But is that really the same "rules" as simply asking "to the best of your knowledge right now, that is the probability the coin was heads"? By summing wins and losses you have added some kind of "memory" to the experiment.

That is a fair objection. You can get rid of the memory by offering at most a single wager. In that case it depends on the rules for offering the wager. If the wager is only offered on Monday, then the credence of heads is 0.5. If the wager is offered only on Tuesday then the credence is 0. If you have some other more complicated scheme for offering the wager then the credence depends on your scheme.

However, if your scheme for the wager is to only offer it on Monday, then what is the meaning of asking the credence on Tuesday in the event of tails? As the problem is described, it is hard to justify any wager other than the simple immediate payout one.

 

[Filip Larsen]

If probabilities don't relate to betting (outcomes in terms of relative frequency), then what are they?

I am trying to argue that the game version of the experiment is a different experiment than the original one (as it is described in the Wikipedia link at least). There is no mentioning of betting or sums of wins/losses in the original version, and I argue that just because its a winning strategy for her to answer tails in some game version of the experiment it does not by itself imply that the answer the question "What is your credence now for the proposition that the coin landed heads?" has the best answer of 1/3. I here stress that I assume that this is the point of the game-argument is trying to make, namely to say that since there is a winning strategy saying tails then tails must be more likely than heads. If that was not yours or Dales intention then I may have misunderstood what argument you are trying to make.

Going back to the original experiment, the key point for me is that she is asked the same question with no change in her knowledge from even before the experiments starts (as we all seem to agree on). So why would she believe P(heads) = 1/3? If I toss a fair coin and ask you to estimate P(head) why would it matter I ask you once more? I mean, you would not say P(head) = 1/2 the first time I ask and then suddenly say P(head) = 1/3 just because I ask again (even if you knew in advance that I would potentially ask twice), so what is the difference in knowledge between you and the sleeping beauty that should make her claim P(head) = 1/3?

 

[Dale]

I am trying to argue that the game version of the experiment is a different experiment than the original one (as it is described in the Wikipedia link at least). There is no mentioning of betting or sums of wins/losses in the original version,

It is the same experiment. The experiment is asking about her credence. Credence is defined in terms of betting.

 

[Filip Larsen]

That is the usual operational definition of credence.

That is also how I understand credence.

 

[PeroK]

I am trying to argue that the game version of the experiment is a different experiment than the original one (as it is described in the Wikipedia link at least). There is no mentioning of betting or sums of wins/losses in the original version, and I argue that just because its a winning strategy for her to answer tails in some game version of the experiment it does not by itself imply that the answer the question "What is your credence now for the proposition that the coin landed heads?" has the best answer of 1/3. I here stress that I assume that this is the point of the game-argument is trying to make, namely to say that since there is a winning strategy saying tails then tails must be more likely than heads. If that was not yours or Dales intention then I may have misunderstood what argument you are trying to make.

Going back to the original experiment, the key point for me is that she is asked the same question with no change in her knowledge from even before the experiments starts (as we all seem to agree on). So why would she believe P(heads) = 1/3? If I toss a fair coin and ask you to estimate P(head) why would it matter I ask you once more? I mean, you would not say P(head) = 1/2 the first time I ask and then suddenly say P(head) = 1/3 just because I ask again (even if you knew in advance that I would potentially ask twice), so what is the difference in knowledge between you and the sleeping beauty that should make her claim P(head) = 1/3?

So, in your view "credence" and "probability" have no relationship to a hypothetical game or bet based on that credence? This is in direct disagreement with mainstream statistical and probability theory.

In this case, you and I might both be sleepers and I might try to persuade you that tails has probability $2/3$, but you stick to $1/2$, regardless of whether you lose money (compared to me) on this belief.

At this point, perhaps I cannot prove that my version of probability theory is right and yours is wrong. But, I do claim that my version of probability theory would be more successful than yours if they were put to the test!

 

[Filip Larsen]

So, in your view "credence" and "probability" have no relationship to a hypothetical game or bet based on that credence? This is in direct disagreement with mainstream statistical and probability theory.

So you and Dale are saying, that credence is to be understood as an equivalent question to the sleeping beauty along the line "given all you know about this experiment are you willing to bet that the coin came up on tails" i.e. the question she is asked each day she is awoken is for her to be understood to have consequences even if she herself cannot remember them?

 

[Dale]

So you and Dale are saying, that credence is to be understood as an equivalent question to the sleeping beauty along the line "given all you know about this experiment are you willing to bet that the coin came up on tails"

Yes. I think that an attempt to introduce more complicated wagers is not justified by the description of the scenario.

 

[PeroK]

So you and Dale are saying, that credence is to be understood as an equivalent question to the sleeping beauty along the line "given all you know about this experiment are you willing to bet that the coin came up on tails" i.e. the question she is asked each day she is awoken is for her to be understood to have consequences even if she herself cannot remember them?

Yes, and I would give a full analysis here.

We have X, who runs the experiment and S, who is the sleeper.

At the start of the experiment the coin is tossed, but no one looks at it. The credences that it is Heads should be:

X = 1/2
S = 1/2

Now, we run the experiment:

Suppose the coin is Heads. This means S gets woken only once. The credences are:

X = 1
S = 1/3

Suppose the coin is Tails. This means S gets woken twice:

First time:

X = 0
S = 1/3

Second time:

X = 0
S = 1/3

The logic is that S cannot distinguish between these three scenarios, which are all equally likely. Note that S can give these answers in advance (before they are woken). There is no new information. The information given at the outset was enough to predict that their credence would always be $1/3$.

Note that S is "correct" in the sense that they give Heads only a 1/3 probability each time and it is Heads 1/3 of the time. They are not using any information about the state of the game, only information about the rules of the game (which was available at the outset). So, no new information.

A sleeper who gives Heads a probability of 1/2 each time is wrong in the sense that they are choosing to ignore the information that they have about the rules of the game.

 

[Moes]

The argument for 50-50 is the one that you have given: it must be 50-50 and no rational person would disagree.

The argument for a third is to do the calculations and find that the answer is one third. There are similar problems where the answer is one third by the same calculations. The 50-50 position, however, is that you are not allowed to do calculations or use mathematics in this specific problem.

This is based on the dubious logic that the sleeping beauty is a princess, not a mathematician, and not capable of mathemetical calculations.

We had a long thread on this a few years ago that turned rather nasty and almost led to my leaving PF.

I wouldn’t say you are not allowed to do calculations just because she is not a mathematician rather my claim is the calculations just don’t apply here or your just using the wrong ones. The answer to the problem wouldn’t change whether she was a mathematician or not.

I am trying to argue that the game version of the experiment is a different experiment than the original one (as it is described in the Wikipedia link at least). There is no mentioning of betting or sums of wins/losses in the original version, and I argue that just because its a winning strategy for her to answer tails in some game version of the experiment it does not by itself imply that the answer the question "What is your credence now for the proposition that the coin landed heads?" has the best answer of 1/3. I here stress that I assume that this is the point of the game-argument is trying to make, namely to say that since there is a winning strategy saying tails then tails must be more likely than heads. If that was not yours or Dales intention then I may have misunderstood what argument you are trying to make.

Going back to the original experiment, the key point for me is that she is asked the same question with no change in her knowledge from even before the experiments starts (as we all seem to agree on). So why would she believe P(heads) = 1/3? If I toss a fair coin and ask you to estimate P(head) why would it matter I ask you once more? I mean, you would not say P(head) = 1/2 the first time I ask and then suddenly say P(head) = 1/3 just because I ask again (even if you knew in advance that I would potentially ask twice), so what is the difference in knowledge between you and the sleeping beauty that should make her claim P(head) = 1/3?

This is basically my claim. I still don’t see how anyone can argue with this. Using the practical betting scenario: Say she was betting 10 dollars on correctly guessing heads or tails. If she knew she was going to be put into this bet every time she woke up then I understand she should choose tails. This is because choosing tails gives her a 50/50 chance of winning 20 dollars or losing just 10. Whereas if she chooses heads it’s vise versa. However this doesn’t show you anything at all about what her credence should be which is the question at hand. (More money doesn’t up the probability of heads or tails.)
(What if she is asked to put one bet on whether she thinks she will gain or lose money with this whole experiment? The chances are surely 50/50 here.)

When she is in the experiment she only knows of being ask into this bet once. So to properly calculate what her credence should be, is to ask if she is told she will only be placing a bet on one awakening whether the coin landed heads or tails. So In this case there is no advantage for her to choose heads or tails when she is woken the chances of her winning or losing the bet are simply 50/50.

I don’t see anyone getting around this. I’m only hearing arguments about what exactly is the question at hand. And when the word “credence” is used I don’t see any room for argument her credence is simply 50/50 she could not be any more sure that the coin landed either heads or tails.

 

[Filip Larsen]

In order to understand the consequences of how the word "credence" in the question affects the sleepers analysis, please consider a variation of the experiment where all is as original except that no matter what the coin lands on, the experimenter will only ask the sleeper the question once. If the coin lands on tails the experimenter will choose at random with day (Monday or Tuesday) he will ask and the other day he will just say "sorry, no question today". The sleeper will know this rule in advance, but when awaken will of course still not remember if he has been awaken or asked before (which means the rules of this variant could equivalently be that the sleeper is only awoken once no matter how the coin lands). According how you define credence, I would think the sleeper should now believe that both heads and tails are equally likely because if he were to bet on either he can expect to come out of the experiment at zero win on average. Is this correct?

If correct I think this way of considering credence to be a measure of probability is "broken" for this experiment (which is no doubt formulated in this way to bring out such conflict). At least it sounds very paradoxical to me that by promising not to repeat a question later that, if asked, is guaranteed to yield same answer, you can somehow affect that answer.

 

[PeroK]

This is basically my claim. I still don’t see how anyone can argue with this. Using the practical betting scenario: Say she was betting 10 dollars on correctly guessing heads or tails. If she knew she was going to be put into this bet every time she woke up then I understand she should choose tails. This is because choosing tails gives her a 50/50 chance of winning 20 dollars or losing just 10. Whereas if she chooses heads it’s vise versa. However this doesn’t show you anything at all about what her credence should be which is the question at hand. (More money doesn’t up the probability of heads or tails.)

Winning bets is precisely what probabilities are about. That's fundamental.

 

[PeroK]

If correct I think this way of considering credence to be a measure of probability is "broken" for this experiment (which is no doubt formulated in this way to bring out such conflict).

That's a good point. Either the thirders are right; or, the halfers are right and probability theory is broken.

I'm happy to be a thirder and retain a working probability theory. And, I'll let the halfers sift through the wreckage of what's left of their probability theory.