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.
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."
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 (-:
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.
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...
They are different sets of families. That is the point of this paradox.
This has nothing to do with quantum mechanics.
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.
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.
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.
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.
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
One of the twins is still considered to be born first and therefore older :-)
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.
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.
"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.
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.
> 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.
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.
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 :-).
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.
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.
Are these not different questions?
> 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.
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?".
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.
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.
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?
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.
In conditional probability theory, there exists something called the boy or girl paradox. The paradox shows that the likelihood of an event can be drastically affected by how you know even seemingly innocuous information. I think that 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. This observation while seemingly innocuous has a drastic affect on the distribution which would explain why we see such confounding results. It follows the same logic as this other naturally occurring phenomenon so I think its an avenue at least worth exploring.
An example of the paradox would be if I told you that my neighbor has 2 children and at least one of them is a boy. If I asked you what the probability that their other child is a girl, normally the result would be 2 / 3, but if I know one child is a boy because I spot him in front of a tree then the odds his other child is a girl drops to 1 / 2.
[0]: https://en.wikipedia.org/wiki/Boy_or_Girl_paradox