Thank you, Mr. Kasper, for your thorough reply. Because all of this is new to me, I feel rather as I did the time I spent an hour on a tennis court with a friend who had won a tennis scholarship to college. Having no real tennis ability myself, I felt I was wasting his time; I appreciated that he’d agreed to play with me for that hour.

As I began to grasp the reasoning, I decided that each time you state the chance that the coin is heads, you are stating a fact. I asked myself what that means. I imagined the following:

I encounter you after you’ve spent two months traveling the world. You address me as follows:

“During my first month, I happened upon one hundred men who told me—each of them—that he had just flipped a coin twice. In each case, I asked, ‘Was at least one of the results heads?’ Each man said yes, and I knew that, in each case, the probability was 1/3 that both flips had been heads.

“In my second month, I again happened upon one hundred men who told me—each of them—that he had just flipped a coin twice. Each added, ‘One of the results was heads; I don’t remember what the other was.’ I knew that, in each case, the probability was 1/2 that both flips had been heads.

“Just as I was about to return home, I was approached by a man who had video recordings of the coin flips that those two hundred men had mentioned. In watching the recordings, I learned that both flips had been heads in fifty of the first one hundred cases and that, likewise, both flips had been heads in fifty of the second one hundred cases.”

In considering that, Mr. Kasper, I imagined the following exchange, which you may imagine as taking place between you and me. I speak first.

“My dog is in that box.”

“Is that a fact?”

“Yes.”

“In saying it’s a fact, you mean what?”

“I mean I regard it as true.”

“Which means what?”

“Which means I can imagine events that culminate in my saying, ‘I seem to have been mistaken; my dog wasn’t in that box.’”

“For example.”

“You walk over to the box and remove its lid, and I see my dog is not in it.”

“Maybe the dog disappeared—vanished into thin air—while I was walking over to the box.”

“That’s a possibility I wouldn’t be able to rule out; but because it would seem to me unlikely, I would say, ‘I seem to have been mistaken; my dog wasn’t in the box.’”

“How much is 189 plus 76?”

“To tell you that, I would have to get a pencil and paper and add them.”

“Please get a pencil and paper and add them; then tell me the result.”

“I’ve just done as you requested. Using a pencil and paper, I’ve added those two numbers. The result is 265.”

“189 + 76 = 265.”

“Correct.”

“Is that a fact?”

“Yes.”

“Please add them again.”

“I’ve just done as you requested. Using my pencil and paper, I’ve added those numbers a second time. I seem to have been mistaken. The result is 255.”

“Are you sure?”

“Well—”

“Please add them again.”

“I’ve just done as you requested. With my pencil and paper, I’ve added the numbers a third time.”

“And?”

“I was right the first time. The sum is 265.”

“Is that a fact?”

“That the sum is 265?”

“Yes.”

“I would say yes. It’s a fact.”

“How much is two plus two?”

“Four.”

“Did you use your pencil and paper to determine that?”

“No.”

“You used your pencil and paper to add 189 and 76 but not to add two and two.”

“That’s right.”

“Is there any sequence of events that could culminate in your saying, ‘I seem to have been mistaken; two plus two is not four.’”

“No.”

“Is it a fact?”

“That two plus two is four?”

“Yes.”

"Yes. It's a fact."

“In saying that, you mean what?”

“—I don’t know.”

Thank you again.

Thank you for the reply, Mr. Kasper.

Let me try this. You come upon a man who, as you watch, flips a 50-50 coin. He catches and covers it; that is, the result of the flip is not known. I, who have been standing there, present you the following question:

"What is the chance the coin is heads?"

That's Question A. What is your answer?

The next day, you come upon a different man, who, as you watch, flips a 50-50 coin. Again, he catches it; again, the result is not revealed. I, who have been standing there, address you as follows:

"Just before you arrived, that man flipped that same coin; it came up heads. What is the chance it is now heads?"

That's Question B. What is your answer?

If you and I were having this discussion in person, I would pause here, to allow you to answer Questions A and B. Because this is the internet, where I don't know how many opportunities you'll have to reply to me, I'll continue.

You come upon a man who is holding a 50-50 coin. I am with him. There is the following exchange:

I (to you, re the man with the coin): This man has just flipped this coin two times.

You: What were the results?

I: One of the results was heads. I don’t remember what the other was.

Question C: What is the chance the other was heads?

Let’s step over Question C (though I'll appreciate your answering it). After I tell you that one of the results was heads but that I don't remember what the other was, you say:

"Which do you remember, the first or the second?"

I reply, "I don’t remember that either."

Question D: What is the chance the other was heads?

Let me try another scenario. A woman says, "I have two children." You respond, "What are their sexes?" She says, "At least one of them is a boy. The other was kidnapped before I was informed of its sex." You're saying that the chance that the kidnapped child is a boy is one out of three, not out of two? To repeat: That's what I gather from the present post, near the beginning of which is the following:

In the correct version of this story, the mathematician says "I have two children", and you ask, "Is at least one a boy?", and she answers "Yes". Then the probability is 1/3 that they are both boys.

Then that I will try on next time.

Just one thing: both you and gwillen mentioned "voting on your example"

I didn't think the point of giving an example was to allow for it being voted. An example is an example. Sure, one may like or dislike it, and think it more or less good. But why is the goal of voting an example important? For me, what matters is creating your own example, and helping those who put theirs.

Advice is 99,9% of time not a 1 bit thing that can be summarized in thumbs down or up.

Agreed this is a good idea. Will do it next time. Though I urge you to consider that the first description people read will shape their kind of response afterwards, so sometimes the "first mover advantage" is good for establishing, tacitly, what kinds of remarks and format of response will be used. (which is a mild instance of what I defend in the solved one)

Though it has some unfairness to it, which may be corrected by separating "expected form" from "first example". Shall do so next time.

I've read a pretty good resolution to the two envelopes problem.

You have to have some prior distribution over the possible amounts of money in the envelopes.

The expected value of switching is equal to 1/2xP(I have the envelope with more money | I opened an envelope containing x dollars) + 2xP(I have the envelope with less money | I opened an envelope containing x dollars). This means that, once you know what x is, if you think that you have less than a 2/3 chance of having the envelope with more money, you should switch.

According to what I read, if your prior is such that there is no finite X for which you would decide that you have less than a 2/3 chance of having the envelope with more money, then your expected value for x, before you learned what it was, was infinite - and if you were expecting an infinite amount of money, then of course any finite value is disappointing. (Note that having a proper prior doesn't keep you from expecting an infinite value for x). So it doesn't matter what you actually find in the envelope, once you find out what it is, you should switch - but there's no reason to switch before you open at least one envelope.

Here's a solution to a more general version of the problem:

Let's say that the red envelope contains N times as much money as the blue envelope with probability p, and the blue envelope contains N times as much money as the red envelope with probability (1 - p).

Without loss of generality, N is at least 1.

If N = 1, both envelopes contain the same amount, and there is no point in switching.

If N > 1, let the variable s represent the smaller amount of money between the amounts of the two envelopes. So one envelope contains s, and the other envelope contains Ns.

Scenario 1: The blue envelope contains s, and the red envelope contains Ns. This scenario occurs with probability p.

Scenario 2: The red envelope contains s, and the blue envelope contains Ns. This scenario occurs with probability (1 - p).

Assume s is not less than 0.

The expected amount of money in the blue envelope, E(B) = sp + Ns(1 - p) = s(p + N - Np).

The expected amount of money in the red envelope, E(R) = s(1 - p) + Nsp = s(1 - p + Np).

If s = 0, both envelopes contain the same amount, and there is no point in switching.

If s > 0, compare the expectations of the amounts of money in the envelopes in terms of s:

E(B) > E(R) when

s(p + N - Np) > s(1 - p + Np)

p + N - Np > 1 - p + Np

2p - 2Np > 1 - N

2p(1 - N) > 1 - N

2p < 1, because (1 - N) < 0

p < 1/2

Similarly, E(B) < E(R) when p > 1/2, and E(B) = E(R) when p = 1/2.

The ratios of the expected amounts of money in each envelope depend only on p when s > 0 and N > 1.

Both envelopes contain the same amount of money when s = 0, N = 1, or both.

So if your degree of belief that the red envelope contains more money than the blue envelope is the same as your degree of belief that the blue envelope contains more money than the red envelope, don't bother switching, unless you need to kill time (which you already knew intuitively, but Q.E.D.)

I got the idea of using the variable s to represent the smaller of the amounts in the two envelopes from R Falk 2008: "The Unrelenting Exchange Paradox".

[Political "gaffe" stories] are completely information-free news events, and they absolutely dominate political news coverage and analysis. It's like asking your doctor if the X-rays show a tumor, and all he'll talk about is how stupid the radiologist's haircut looks. . . . ["Blast"] stories are. . . just as content-free as the "gaffe" stories. But they are popular for the same reason: There's a petty, tribal satisfaction in seeing a member of our team really put the other team in their place. And there's a rush of outrage adrenaline when the other team says something mean about us. So, instead of covering pending legislation or the impact it could have on your life, the news media covers the dick-measuring contest.

-David Wong, 5 Ways to Spot a B.S. Political Story in Under 10 Seconds

It's weird how proud people are of not learning math when the same arguments apply to learning to play music, cook, or speak a foreign language.

http://xkcd.com/1050/