I A variation of the sleeping beauty problem

[Dale]

I agree that this scenario is pretty clearly 1/3. It seems that this is just a straightforward conditional probability exercise.

Interestingly, a glance at the truth table shows that anyone that is not asked knows that the coin was heads with (conditional) probability 1. That is kind of obvious, but it hadn't struck me until I did the table. One possible inequivalence is that in the setting of repeated experiments there would be participants with this complete certainty. That never happens in the original.

On the other hand since such individuals are never asked about the probability, the fact that they exist doesn't necessarily make the scenario inequivalent with respect to probability. But I think that it at least makes the claim of equivalence require some solid justification. On what basis can equivalence be claimed and/or rejected? I really don't know.

 

[jbergman]

Actually, I am not sure they are inequivalent. Rosenthal, claims they are and analyzes this exact variant in A Mathematical Analysis of the Sleeping Beauty Problem. I agree with his analysis there and the consensus that the answer is 1/3.

Just to post a slight translation of Rosenthal's analysis here, since it is fairly convincing for this variant of the problem. He actually introduces two coins one is a nickel that we flip first and then we flip a dime if the nickel was a heads to decide whether to interview SB on Monday or Tuesday. Then we have the following:

P(Nickel = Heads and Asked On Monday) = P(Nickel = Heads and Dime = Heads) = 1/4
P(Asked On Monday) = P(Nickel = Tail or Heads and Dime = Heads) = P(Nickel = Tail) + P(Nickel = Heads and Dime = Heads) = 3/4

Therefore P(Nickel = Heads | Asked On Monday) = P(Nickel = Heads and Asked On Monday) / P(Asked On Monday) = 1/3.

 

[PeroK]

There is another "paradox" in elementary probability theory. Suppose we toss three coins. They can't all be different, so we can find two that are the same. The third is then the same or different with equal probability. Hence, the probability they are all the same must be 1/2.

But, if we run a computer simulation, we find that the probability they are all the same is 1/4.

We can draw two possible conclusions:

Probability theory is ambiguous and both 1/2 and 1/4 are valid answers.

The answer of 1/4 must be correct, by experiment. The 1/2 argument must be flawed.

The question in the OP can be resolved by computer simulation. The arguments are only important in terms of understanding the answer; not in establishing it. The answer is 1/3, whether we like it or not. And if someone produces an argument that the probability is not 1/3, then (as Feynman would have said) it doesn't matter what their name is, and it doesn't matter how clever they are, it's wrong!

 

[PeterDonis]

The arguments are only important in terms of understanding the answer; not in establishing it.

I'm not so sure. Suppose I simulate a million trials and all three simulated coin flips are the same in 243,596 of them. Is the probability of the outcome 243,596/1,000,000? Or is it 1/4? Experiment can't tell me the answer to that question; only theory can.

This experiment can of course rule out 1/2 as a valid answer, at least to a very high confidence level. But it can't tell me that the answer is exactly 1/4, instead of whatever ratio the experiment yielded. I need a theoretical argument to predict the theoretical ratio of 1/4, to which I can then compare what I see in the experiment and decide whether I accept the theory or need to modify it.

 

[fresh_42]

Before we run again in a complete subthread which would be hard to separate among those many posts I'd like to ask you to either stay on topic or create a new thread with this new topic. Thank you.

 

[PeroK]

Before we run again in a complete subthread which would be hard to separate among those many posts I'd like to ask you to either stay on topic or create a new thread with this new topic. Thank you.

Okay, but I'm not sure how much more we can squeeze out of this thread.

 

[PeroK]

This experiment can of course rule out 1/2 as a valid answer, at least to a very high confidence level.

Not if it's the sleeping beauty problem it can't!

 

[fresh_42]

Okay, but I'm not sure how much more we can squeeze out of this thread.

I think that you are very right here, so I will close the discussion.
Please contact me in case you still have something important to say and I closed it too early.

As for the latest development (posts #53 f.):

"Significance of probability measurements and distribution of error margins."​

seems to be a debatable subject among applied mathematicians and physicists.