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I think the problem with QM isn't the fact that the probabilities are confusingly calculated. It's that there's no underlying mechanism.

Imagine I flip a fair coin. The odds of you guessing it correctly are 50%. In your mind, you know it's probabilistic. Simultaneously, you imagine that if you had more information you could get it right more than 50% of the time. Maybe if you new the force or the initial velocity, that'd help. The thing with QM, there's no more information to get. You can't get better than 50%, ever. It's basically magic. The problem with any deterministic model is that it adds more complexity than just assuming magic and we can't test any of the predictions of current deterministic models so why bother.

That's a key point - classical randomness is still causal.

QM effectively seems to be non-causal - or at least, much less causal.

Tidying this away under the heading of "It's probabilistic, let's just do the math and not worry about why" seems to be ignoring some very basic questions.

I can't seriously imagine Newton saying "Well, the Moon stays in the sky instead of falling down - and that's frankly a bit weird, but I'm not going to wonder why because it's just a mystery and anyway, I can predict its orbit with these epicycles."

> I can't seriously imagine Newton saying ..

But he did. He called it Gravity.

He didn't let figuring out "how gravity works" hold him back. He just figured out the equations that predict its effects based on observation. The unknowns happened to conveniently coalesce around a single constant G, but that doesn't make them "known".

[IANAP - do we have any convincing popularly accessible models for how gravity works even today?]

Anyway, that's kind of similar to going "oh I don't understand the subatomic mechanism that is causing this interference pattern, but the pattern itself is consistent so lets call that P!".

Unless I completely sped past the point you were trying to make - which is completely possible at this subatomic level of reasoning (-:

Actually, both you and GP got it wrong. There is no need to speculate what Newton was thinking, it was all recorded.

Newton was looking for methods to manipulate (push, pull) objects at a distance without touching them. He had strong ideas about the occult and what we today call black magic, in addition to being an alchemist. He devised that the Earth was manipulating the untouched Moon (capitalized, proper noun) and sought to explain the influence of the Earth on the Moon.

LOL is that for real! Please please can you give me a couple of good citations for that?

Interestingly, I thought that I could simply point you towards Wikipedia [1] and call it a day. However, obviously Newton's viewpoints were controversial in his day and remain so. The controversy is hinted at in Wikipedia, without details. Even the "History of gravitational theory" [2] article completely eliminates this aspect and the "Newton's law of universal gravitation" concentrates completely on the dispute with Hooke.

Many science-related articles on Wikipedia suffer from good-meaning but over-zealous gatekeeping of religious influences on science. So I'll just recommend to you to read a biography or three on Newton. His entire story is fascinating and we can see how tiny events in his childhood (death of father, influence of priest uncle) have affected the history of physics.

[1] https://en.wikipedia.org/wiki/Isaac_Newton [2] https://en.wikipedia.org/wiki/History_of_gravitational_theor... [3] https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gr...

No, that is not what this says. What this says is that there is ambiguity in the description of the situation. Is the question about the probability of a second girl in the set of all families with 2 children and at least one boy, or is the question about the probability of a second girl in the set of all families with 2 children given that we know one of them is a boy.

They are different sets of families. That is the point of this paradox.

This has nothing to do with quantum mechanics.

Say my wife tells me that the neighbor has 2 kids and one of them is a boy what is the likelihood that their other child is a girl?

Given what I know about the current situation, the probability that their other child is a girl is 66.666%

If I look out the window to that neighbor's yard and see a boy playing outside, the probability that their other child is a girl drops to 50%.

How I know the information about on data point one affected the likelihood of an unrelated one. What seems like identical knowledge on the outside, changes outcomes based on how I accrued that knowledge. If that doesn't mimic the double-slit experiment, I don't know what does.

No, that's not what this is about.

Probability means this: "in X tries, how likely is it for Y to happen". The ambiguity here is this: what is the set of X? Is the set of all families with two children at least one boy, or the set of all families with 2 children of which you have determined one is a boy.

Those are different sets. It doesn't matter how you discovered the information. The apparent paradox is all grammar and set description.

Think of it this way - you want to test this out and actually run this experiment? Great! What's the experiment? Are you starting with all families with 2 children, or all families which 2 children at least one boy.

>> How I know the information about on data point one affected the likelihood of an unrelated one

No, it doesn't.

It doesn't make sense to me that in order to get the probability of 2/3, you implicitly consider "a boy and a girl" to be different from "a girl and a boy".

Somehow, you're saying that seeing the boy means you have ruled out the possibility of "a girl and a boy". But the ordering is in no way related to some arbitrary observation. It's just lexical.

Here's the math, I will break it down based on Boy/Girl and seen.

For the first case here are the different combinations for two children based on sex (let's say left denotes older and right denotes younger to distinguish the difference between girl and a boy and boy and a girl.)

B B

B G

G B

G G

These outcomes are each equally likely, but since we know that one has to be a boy we can eliminate the (G G) case leaving us with two chances out of three that the other child is a girl.

The other case can be viewed as follows (S) denotes seen.

B(S) B

B B(S)

B(S) G

B G(S)

G(S) B

G B(S)

G(S) G

G G(S)

I can eliminate all cases where I see a girl out the window, leaving me with the following equally likely cases:

The neighbor has two boys and I see the older boy. The neighbor has two boys and I see the younger boy. The neighbor has an older boy whom I saw and younger girl. The neighbor has an older girl and a younger boy whom I saw. The neighbor has a girl in 50% of the remaining possible cases.

Suppose they are twins and therefore there is no older and younger. Does that change the probability from 2/3 in the first case?

If you assume that the siblings can be differentiated (thus ordered), it makes no difference. You can for example order them taking their names alphabetically, or by weight, or by location at a given time, etc...

If differentiation is not possible, then they are the exact same person [1], so the parents have one child and it must be boy then ;)

[1] https://en.wikipedia.org/wiki/Identity_of_indiscernibles

Ignoring the increased probability that twins are of the same gender, no. Regardless, the chance of having two children of different genders is twice that of having two boys. If you split the set into 'two boys, a boy and a girl, and two girls', you must accept that the 'a boy and a girl' case is twice as likely as the first and last options.

I guess this is one of a few place to answer mu.

One of the twins is still considered to be born first and therefore older :-)

I wrote a program and successfully got the right answers experimentally for each case, so I'm not doubting the answers, but I don't buy the explanation.

While you may argue that one twin is always born first, this is an idealized mathematical problem, so it can't depend on particular details of physical reality in that way. There is nothing logically preventing them from being the same age, so it can't be pivotal to solving the problem.

And in fact, I experimentally got the right result without any reference to ages.

So, you don't actually have to regard BG and GB as separate cases if you don't want to.

People like to write the outcome space as ordered pairs BB, BG, GB, GG (ordered by age or whatever else), because this has the advantage that all four outcomes are equiprobable, so you can calculate probabilities just by counting.

If you prefer to regard the outcome space as unordered pairs {B,B}, {B,G}, {G,G} this is absolutely fine: you just have to bear in mind that the prior probability of {B,G} is 1/2, whereas it's 1/4 for the other pairs, so a tiny bit more calculation might be required.

P({B,G}| not {G, G}) = P({B,G})/(P{B,B} + P{B,G}) = (1/2)/((1/2) + (1/4)) = 2/3 as with the other method.

I'm having a fundamental problem understanding why the paradox is modeled the way it is.

"Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?"

For some reason, it's expanded as a choice from {BB, BG, GB, GG}. But why on Earth does the order of children matter in this question? The way I see it, it should be modeled as {both-boys, boy-and-girl, both-girls}; by telling me that one of the kids is a boy, you eliminated one possibility out of three, leaving us with 1/2 as an answer.

Let's say I flipped two coins. I tell you I did this. You ask me, "Did heads come up at least once?" I answer yes.

What are the odds that the other coin was tails? 2/3, because there are three scenarios (HH, HT, TH) that satisfy my first answer to you.

If you flip a pair of coins a few thousand times, and search the flip pairs for all the ones that had a heads among them, you'll find a tails in those pairs 2/3 of the time, and a pair of heads 1/3 of the time.

Same with the girl-boy problem.

Now if you ask me if the left coin came up heads—and I say yes—that doesn't give you any information whatsoever on the right coin. And if you search through the thousands of pair flips looking for an "H" in the left column, the right column will 50/50 have an H or a T.

That's a good explanation.

> Now if you ask me if the left coin came up heads—and I say yes—that doesn't give you any information whatsoever on the right coin.

Exactly. "At least one" gives information about the whole set, implicitly giving information about the second one whereas "the first one is X" gives no information about the second one.

Because if you select a random family from the set of familes with two children, the 4 permutations (BB,BG,GB,GG) are equally likely, at 25% each, so it's the simplest way to model it. If you model it in your way with 3 possible scenarios you'll find that the probabilities are BB: 25%, one of each: 50%, GG: 25%, and dealing with those differing probabilities is harder than just splitung the one of each scenario into two.

Try it with 2 coins flipped at the same time and you'll see it's the case. You'll get 2 unmatching coins about twice as often as you get either HH or TT.

I read the wikipedia article and then it clicked for me. Here is the intuition I got:

If you know one of the kids is a boy then there is a 2/3 chance of there being a girl kid. I will assume you are happy with the reasoning behind this (3 options BG GB BB, two of which yield a girl).

Now why does spotting one of the boys change the odds? You already knew they had a boy right?

Well the boy being spotted conveys some information!

In particular you DIDN'T see a girl. You could have seen one, it would be possible right... but it didn't happen.

That fact means it is a bit more likely that they have two boys. How much is that bit, well 1/6th to be precise. So 1/3+1/6 = 1/2 chance they have two boys. And 2/3-1/6 = 1/2 chance they have two girls.

That gives me the intuition and makes this no longer a paradox intuitively. Although it was never a paradox logically.

CAREFUL:

The conditions under which the boy was 'revealed' to you matter.

If it was random 50-50 chance of seeing the girl or the boy then the above applies.

If the boy was picked, i.e. you said "ah you have at least 1 boy, show him to me". Then no information has been conveyed.

If something else situational (e.g. girls birthday party and all the girls are inside in the room, and boys are playing soccer outside [sorry for stereotyping but my imagination isn't so good] and so you see him) there may be a non 50-50 and non 100-0 probability then there are some more calculations to do :-).

>The way I see it, it should be modeled as {both-boys, boy-and-girl, both-girls}; by telling me that one of the kids is a boy, you eliminated one possibility out of three, leaving us with 1/2 as an answer.

This would be like saying "When I buy a lottery ticket, the only outcomes are {lose lottery, win lottery}; therefore, I have a 50% chance of winning the lottery."

The difference is that boy-girl is twice as likely to occur of those three choices.

It's not the ordering that matters, it's simply that the odds of boy-and-girl are twice the odds of boy-and-boy.

I think the way you stated the first problem is ambiguous; the answer could be 1/2 or 2/3. After all, maybe you only have that information because you've met just one of your neighbour's children who happens to be a boy.

The Wikipedia article does a great job of being explicit in the differences between the 1:2 and 2:3 cases.

Unfortunately articles pop up every now and then claiming 2:3 is the correct answer (since it's counterintuitive and therefore interesting), while losing the conditions necessary for 2:3 in the translation of the problem description from mathematical to plain English.

Information changes the question asked. You don't state the questions. Did you mean: neighbor has 2 children, we don't know the sex, what is the probability one or more is a girl? Next question is: neighbor has 1 child of unknown sex, odds of it being a girl?

Are these not different questions?

This is actually a very interesting musing, and not quite correct.

> observation of the atoms passing through the slit is affecting the search space for the other atoms in such a way that we currently do not understand

Observation is the key word here. Actually, you don't need to observe the atoms to affect the pattern on the screen - though its the easiest way to do so in the double-slit experiment. All you need is the possibility of observing or distinguishing between two events to destroy the interference between them. Note that possibility of observing is a much weaker condition than observation itself. Physicists have, over the years, done many experiments to clarify these issues. Essentially, if you believe that causality only flows forward through time, then what you are saying is not correct.

If it were just a probability paradox, Bell's theorem wouldn't be a thing. You can win certain coordination games more often using quantum correlations than is possible with classical correlations.

That being said, I have complained about specific papers and articles making this exact mistake. For example, explanations of delayed choice eraser experiments are almost always confusing correlation or post-selection for causation ( http://algassert.com/post/1720 ). But Bell tests are not of that form, which is why the first thing you should ask of any proposed interpretation is "How do you deal with Bell tests?".

It's more like seeing one boy actually impacts the gender of the second child, which is why the quantum experiment is so mind boggling. It doesn't seem to have any resemblance with classical physics and only makes sense in quantum physics.

That's exactly what the paradox states. Seeing one boy does affect the gender of the second child.

No it doesn't. In the 2/3 case, you're given information about the whole set, not just one member. Instead of "seeing one boy" affecting the gender of the other child, it's "seeing there are more than 0 boys" affecting the gender of one of the children.

Here's another example: If I flipped 100 coins, and told you there were 99 heads, what's the likelihood the other 1 was tails? The answer is 100/101, because there are 100 ways the one tail could happen, while only one way zero tails can happen. But if you're only told that the first 99 are heads, the answer is 1/2. You know the last flip has exactly the same odds as all the other ones -- the trials are independent. The difference is in one all the flips were included in the results (and the question isn't about any one specific flip), while in the other you had no information about the last flip (which the question was about).

Another way of looking at it: In the first example, the question was asking about the sum of 100 trials and you were given information about 99 of those 100 trials. You were given 99/100s of the information you need to answer the question. In the second example, the question was asking about the last trial and you were given information about 99 unrelated trials beforehand. You were given none of the information you need to answer the question.

It's not a paradox if you put it in less confusing language.

There are 4 possibilities: GG GB BG BB

GG is impossible so you have GB, BG, & BB. One of them is a B, what's the other. Well that's clearly 2 / 3 G.

The other one situation is completely. It's basically just the chance of a child being a girl.

Is this basically to do with the fact that you know how likely it is to see a boy given any of the combinations? So there's a 33% chance that there are two boys and you see a boy, a 33% chance that there's a boy and girl and you see a boy, and a 33% chance that there's a boy and a girl and you see a girl. Given that you've seen the boy you've eliminated one of the outcomes so the other two outcomes are equally likely?

I'm confused by your comment but I'll try to explain more.

Given one of the children is a boy. The only combinations possible are: GB, BG, & BB. Where G is Girl and B is Boy. The question is about the probability of the other child being a Girl.

When we eliminate one of the B from each combination, we get:

GB, BG, BB -> G, G, B

We can see that 2/3 are G and 1/3 are B.

Does this help?

But are those ordered pairs or sets of two items each?

I think of them as ordered pairs, but would it change anything if they were sets? Order doesn't matter in this question.

Order doesn't matter but it helps make it more clear as why I included the same combination twice.

The boy or girl paradox does appear in physics but not like this. Distinguishable particles behave like "older child is a girl", while indistinguishable particles behave like "at least one child is a girl"

It's a paradox only because how question is put. Your wife says that neighbours have children, and she observed a boy. Then she observed boy again. What difference in probability having neighbours having a girl if you seen a boy once or twice.

You have red and black pebbles otherwise identical. Bag has 2 pebbles, you draw a pebble, it's red. You put it back. What's the probability of bag having black and red pebble? Draw again - red again. What's the probability of bag having black and red pebble? You acquire more data, therefore your estimate changes.

I'm afraid I don't follow. Can you elaborate on how that "paradox" is an analog of the double slit experiment?