“given that all rolls were even” here means “roll a normal 6 sided dice, but throw out all of the sequences that included odd numbers.” The two are not the same, because in the case where odd numbers can be rolled, but they “kill” the sequence it makes situations involving long sequences of rolls much less likely to be included in the dataset at all.
As other comments explain, this is why the paradox emerges. By stealth, the question is actually “A: How long do I have to wait for two 6s in a row, vs B: getting two 6′s, not necessarily in a row, given that I am post selecting in a way that very strongly favors short sequences of rolls”.
Yes, exactly—thank you. It depends on the interpretation of the phrase “given that all rolls were even”. Most ordinary people will assume it means that all the rolls were even, but as you have succinctly explained, that is not what it means in the specialist language of mathematics. It is only when you apply the latter interpretation, that some of the rolls are odd but we throw those out afterwards, that the result becomes at first surprising.
I do find LessWrong a curious place and am not a regular here. You can post something and it will get downvoted as wrong, then someone else comes along and says exactly the same thing and it’s marked as correct. Heh.
Yes, its a bit weird. I was replying because I thought (perhaps getting the wrong end of the stick) that you were confused about what the question was, not (as it seems now) pointing out that the question (in your view) is open to being confused.
In probability theory the phrase “given that” is a very important, and it is (as far as I know) always used in the way used here. [“given that X happens” means “X may or may not happen, but we are thinking about the cases where it does”, which is very different from meaning “X always happens”]
A more common use would be “What is the probability that a person is sick, given that they are visiting a doctor right now?”. This doesn’t mean “everyone in the world is visiting a doctor right now”, it means that the people who are not visiting a doctor right now exist, but we are not talking about them. Similarly, the original post’s imagined world involves cases where odd numbers are rolled, but we are talking about the set without odds. It is weird to think about how proposing a whole set of imaginary situations (odd and even rolls) then talking only about a subset of them (only evens) is NOT the same as initially proposing the smaller set of imaginary events in the first place (your D3 labelled 2,4,6).
But yes, I can definitely see how the phrase “given that”, could be interpreted the other way.
I’m not sure about the off-topic rules here, but how about this:
Why are some of the drinks so expensive, given that all of them are mostly water?
Sometimes we use the phrase “given that” to mean, “considering that”. Here, we do not mean, some of the drinks are not mostly water but we are not talking about them. We mean that literally all the drinks are mostly water.
Jumping in here: the whole point of the paragraph right after defining “A” and “B” was to ensure we were all on the same page. I also don’t understand what you mean by:
Most ordinary people will assume it means that all the rolls were even
and much else of what you’ve written. I tell you I will roll a die until I get two 6s and let you know how many odds I rolled in the process. I then do so secretly and tell you there were 0 odds. All rolls are even. You can now make a probability distribution on the number of rolls I made, and compute its expectation.
“given that all rolls were even” here means “roll a normal 6 sided dice, but throw out all of the sequences that included odd numbers.” The two are not the same, because in the case where odd numbers can be rolled, but they “kill” the sequence it makes situations involving long sequences of rolls much less likely to be included in the dataset at all.
As other comments explain, this is why the paradox emerges. By stealth, the question is actually “A: How long do I have to wait for two 6s in a row, vs B: getting two 6′s, not necessarily in a row, given that I am post selecting in a way that very strongly favors short sequences of rolls”.
Yes, exactly—thank you. It depends on the interpretation of the phrase “given that all rolls were even”. Most ordinary people will assume it means that all the rolls were even, but as you have succinctly explained, that is not what it means in the specialist language of mathematics. It is only when you apply the latter interpretation, that some of the rolls are odd but we throw those out afterwards, that the result becomes at first surprising.
I do find LessWrong a curious place and am not a regular here. You can post something and it will get downvoted as wrong, then someone else comes along and says exactly the same thing and it’s marked as correct. Heh.
Yes, its a bit weird. I was replying because I thought (perhaps getting the wrong end of the stick) that you were confused about what the question was, not (as it seems now) pointing out that the question (in your view) is open to being confused.
In probability theory the phrase “given that” is a very important, and it is (as far as I know) always used in the way used here. [“given that X happens” means “X may or may not happen, but we are thinking about the cases where it does”, which is very different from meaning “X always happens”]
A more common use would be “What is the probability that a person is sick, given that they are visiting a doctor right now?”. This doesn’t mean “everyone in the world is visiting a doctor right now”, it means that the people who are not visiting a doctor right now exist, but we are not talking about them. Similarly, the original post’s imagined world involves cases where odd numbers are rolled, but we are talking about the set without odds. It is weird to think about how proposing a whole set of imaginary situations (odd and even rolls) then talking only about a subset of them (only evens) is NOT the same as initially proposing the smaller set of imaginary events in the first place (your D3 labelled 2,4,6).
But yes, I can definitely see how the phrase “given that”, could be interpreted the other way.
I’m not sure about the off-topic rules here, but how about this:
Why are some of the drinks so expensive, given that all of them are mostly water?
Sometimes we use the phrase “given that” to mean, “considering that”. Here, we do not mean, some of the drinks are not mostly water but we are not talking about them. We mean that literally all the drinks are mostly water.
Jumping in here: the whole point of the paragraph right after defining “A” and “B” was to ensure we were all on the same page. I also don’t understand what you mean by:
and much else of what you’ve written. I tell you I will roll a die until I get two 6s and let you know how many odds I rolled in the process. I then do so secretly and tell you there were 0 odds. All rolls are even. You can now make a probability distribution on the number of rolls I made, and compute its expectation.