6 ms·

By far the most unintuitive paradox for me personally is the one presented here: https://youtu.be/go3xtDdsNQM?t=3m27s https://youtu.be/go3xtDdsNQM?t=3m27s "Mr.

By far the most unintuitive paradox for me personally is the one presented here: https://youtu.be/go3xtDdsNQM?t=3m27s https://youtu.be/go3xtDdsNQM?t=3m27s "Mr. Jones has 2 children. What is the probability he has a girl if he has a boy born on Tuesday?" Somehow knowing the day of the week the boy was born changes the result. It's completely bizarre.

Jenya_ 9y ago
The comments to this video actually say (with proof) that this was an error in the video.
- Houshalter 9y ago
  There seems to be quite a bit of debate about it in the comments and I'm not sure who to believe. At one point someone coded a simulation to test it and the results were as predicted by the video. Even if the video is incorrect, the fact it's so confusing still makes it an interesting paradox.
Retric 9y ago
It's fairly simple. If you flip 2 coins then say whatever the first coin was the the odds if you said H was HH, or HT and if you said T it would be TH, TT. However, if you flip two coins and then say if you got at least one head independently from whatever you flipped then the odds you have 3 options HT, HH, TH with equal odds. So, the question is if the full statement was based on the data or only the truth value of the statement is based on the data. PS: Now assuming it's truth value is based on data. if you look at all options there are 14 gender day combinations per kid and 14 * 14 = 196 gender day combinations in totoal. Only 14 of of those 196 start BT which is then split evenly 7 BTB_, 7 BTG_. However that leaves 196 - 14 other options to consider. 7 * 14 of them Start G, and 6 * 14 of them start with B not on a Tuesday, but out of those you only keep 1/14 as you need BT on the second roll. Now add them up 13B and 14G out of (13 + 14) = 27. Or 13/27 B, and 14/27G.
Chinjut 9y ago
Your problem is that you are thinking there's a "the boy". But there's not a "the boy". Mr. Jones could have two boys. He could have two boys both born on Tuesday, even. The term "the boy" does not denote any particular boy, in that case, and causes you to think about the situation erroneously. If the question were "There's Kid 1 and Kid 2, each independently selected with random gender and birth-day-of-the-week. Out of those cases where Kid 1 is a boy born on Tuesday, what proportion are cases where Kid 2 is a girl?", then the answer would indeed be a straightforward 50%; the status of Kid 1 is entirely independent of the status of Kid 2. But that's not the question. The question is "There's Kid 1 and Kid 2, each independently selected with random gender and birth-day-of-the-week. Out of those cases where at least one (either one, and possibly both) of Kid 1 and Kid 2 is a boy born on Tuesday, what proportion are cases where at least one of Kid 1 and Kid 2 is a girl?". This is very different, and of course just drawing out the possibilities (all 2 * 7 * 2 * 7 equiprobable-by-stipulation choices of gender and birth-day-of-the-week for Kid 1 and Kid 2) and circling which pairs of subsets are the relevant ones for the two questions reveals the difference, the probabilities for either question elementarily calculable in this way by basic counting.
- Shendare 9y ago
  Agreed. He removed the second B2B2 probability annotation as though it were a repeat of the first and inapplicable to the probability set, but that's not the case, and it shouldn't be removed. Apply lower-case to the younger boy in the probability sets and it's clear why. B2b2 is not the same occurrence as b2B2. Even though the day both were born on was "a Tuesday" doesn't mean both probability instances are referring to the exact same event. Except in the case of twins, which is outside the scope of the exercise.
  - deleted 9y ago
    [deleted]
- Chinjut 9y ago
  The 14/27 answer in the video is correct, incidentally. Also, I notice you said "Somehow knowing the day of the week the boy was born changes the result. It's completely bizarre." Remember, though, there's no "the boy". The question "On which day of the week was the boy born? Tell me, I need to know!" does not always have a well-defined answer. Indeed, you'd get the same 14/27 answer even if "Tuesday" in the question "What proportion of two-children families with at least one Tuesday boy have a girl?" was replaced by any other day. And if this seems paradoxically in conflict with the fact that simply asking "What proportion of two-children families with at least one boy have a girl?" has instead the answer 2/3, reflect again upon the fact that some families have two boys born on different days, so that there's no single answer to "On what day was 'the boy' born?". And then just draw out the cases and count. (Specifically, out of the 2 * 7 * 2 * 7 equiprobable cases overall for Kid 1 and Kid 2's genders and days, there are 27 cases where there's at least one Tuesday boy, and 14 cases where there's at least one Tuesday boy and also a girl. There are 3 * 7^2 cases where there's at least one boy, and 2 * 7^2 cases where there's at least one boy and also a girl.) Many of these questions, I think, become clearer if thought of as counting questions instead of as "probability" questions (though it's all the same; the math called "probability" is just the math of various kinds of counting (from simple counting as in this case to complexly weighted continuous measurements, but still ultimately a generalized form of counting). However, despite that equivalence, the concept "probability" has developed all these other distracting connotations, such that psychologically, there can be a useful difference in perspective in switch to explicitly thinking "counting" instead. No one would long dispute that there are 27 cases with at least one Tuesday boy, etc.).
justinpombrio 9y ago
The question is ill-posed: it does not give you enough information to tell the probability. You know what Mr. Jones has told you, but you don't know under what circumstances he would have told you this. Suppose that you ask Mr. Jones weather he has a boy and he says yes. Then the probability that he also has a girl is 2/3. Suppose that you asked Mr. Jones weather he had a boy born on a Tuesday, and he says yes. Then the probability that he has a girl is less than 2/3, because having two boys gives (about) double the chance for one of them to have been born on a Tuesday. However, suppose that you asked Mr. Jones weather he has a boy, and if so what day his eldest boy was born on, and he says "yes, and on Tuesday". Then the probability that he also has a girl is again exactly 2/3. Wikipedia has a detailed explanation: https://en.wikipedia.org/wiki/Boy_or_Girl_paradox https://en.wikipedia.org/wiki/Boy_or_Girl_paradox
- Chinjut 9y ago
  Everything you say after your first paragraph is correct (presuming people always answer questions with "Yes" or "No" honestly), but… No one said anything about "Mr. Jones has told you…", here. There was nothing about asking Mr. Jones a question and him providing an answer according to some process. Rather, the question was simply "Mr. Jones has two children. What is the probability he has a girl if he has a boy born on Tuesday?". There are implicit conventions involved in reading this, but not particularly problematic ones. This implicitly means "Out of all families with two children, at least one of which is a boy born on Tuesday, what proportion have a girl? [Presuming that out of those families, birth gender and day of the week for the two children are all independently uniformly distributed]". And this is a straightforward counting problem. So the wording seems fine and the problem well-posed to me.
  - Retric 9y ago
    There are implicit conventions involved in reading this Explicitly the question adds no such limits. So, abstractly someone could be asking the question without those limits. It's like the difference between infinity and how whatever subset of math you work in defines infinity. And yes there are more than one commonly used definition.
astrocat 9y ago
I'm an idiot, but I'm going to throw my hat in the ring here: The video is wrong. The problem reads: Jones has 2 kids. What is P(he has a girl) given that he has a boy born on a Tuesday. Consider, for a moment, what information we're getting from "boy born on a Tuesday." This is no different than "boy with red hair," or "boy with 5 freckles." The fact that the BOY was born on a tuesday does not change P(day of the week girl was born). Imagine the "boy with 5 freckles" case - let 5 freckles be denoted by F5, six freckles by F6 and so on... would the appropriate calculation include enumerating P(boy F5, boy Fn) for all n? No. The "born on Tuesday" is irrelevant. Thus you have the following scenarios: - one kid is TuesdayBoy and the other is also a boy, born at any time - one kid is TuesdayBoy and the other is a girl, born at any time Out of these options P(Jones has a girl) is a flat out 50%. There is no need to bring in concepts of "which was born first" or enumerate all possible days of the week each child could have been born. Ok... now all the real smartypants here can correct me :)
- Chinjut 9y ago
  There are 2 * 7 * 2 * 7 ways to assign gender and birth-day-of-week to two children. By convention, all are considered equiprobable (this is the same as assuming kids' genders and birth day-of-weeks are independent of each other and of all facts about other kids, and that both genders are equally likely and all 7 days are equally likely for any given kid.) Of these possibilities, 27 are situations where one kid is a Tuesday boy. [Do you dispute this count?] Of those, 14 are situations where one kid is a girl. [Do you dispute this count?] The answer to "What proportion of cases where there is at least one Tuesday boy also have a girl?" is thus 14/27. You have stated by fiat that certain things are irrelevant to certain other things, that certain things have probability 50%, etc, but in doing so, you have not considered the count correctly. You are likely misled by phrasing such as "the boy", when there are families with two boys in which there is no proper referent of "the boy" and no particular answer to question like "Which day was 'the boy' born?".
  - astrocat 9y ago
    ok ok... let me try to get this straight. Just as kind of a mental process for trying to understand whether or not something passes the smell test, I typically try to take the basic premise and turn it up to 11 and see if that still makes sense. In this problem, as you've described it, we're enumerating "ways to assign gender and birth-day-of-week." We can do this because there are a countable number of "days of the week" (so we can map to the integers: 1-6) AND there is also a surjective function of [child] -> [day of the week they were born]. Am I right so far? Now let's replace the set [1-6] with another countable set that also maintains the surjective function. We could say "day in the lunar cycle" (so ~27 options), or better "day of the year" (366 options), for example. Do we now need to consider the 23662366 ways to assign gender and birth-day-of-the-year? Take it further with whatever you want: "birth weight in milligrams" or "number of freckles" (as I previously suggested). All countable things that meet the surjective requirement. This is starting to smell funny, right? So let's take a look at the math. You say there are 2727 ways to configure day+gender, assuming independence for kid 1 (k1) and kid 2 (k2). This represents: (k1 gender options * k1 day of week options) * (k2 gender options * k2 day of week options). Right? I'm with you so far. Then you say "Of these possibilities, 27 are situations where one kid is a Tuesday boy." Hold up. We are given two pieces of information: that one of the kids is a boy, and that particular boy was born on a Tuesday. Let's say the boy is k1 (this is an assignment of enumeration, not of "who came first;" just like Sunday = 1 does not mean that any kid born on a Sunday was born before every kid born on Monday = 2). So now the k1 options are [11] (boy, tuesday), and the total number of options are: [11] * [27] = 14. Of those 14, 7 are girl options. And we're back to a straight 50%. So yes, I dispute the 27 number. It seems like it is arrived at by 2127, minus one for an apparent duplicate. But the 212*7 represents maintaining gender non-specificity for Tuesday boy, which should be incorrect, no? > You have stated by fiat that certain things are irrelevant to certain other things... Yes, but that's what "independent" means, right? You also stated that you're assuming these two things are independent, hence equiprobability. But independence is defined by P(A) = P(A|B). The probability of A is completely unaffected by B. Yet the outcome you arrive at is that P(A) IS affected by B, so the math presented is internally inconsistent. What am I missing here? I'm fascinated by the uncertainty around this little problem.

Jenya_ 9y ago

The comments to this video actually say (with proof) that this was an error in the video.

Houshalter 9y ago
There seems to be quite a bit of debate about it in the comments and I'm not sure who to believe. At one point someone coded a simulation to test it and the results were as predicted by the video. Even if the video is incorrect, the fact it's so confusing still makes it an interesting paradox.

Retric 9y ago

It's fairly simple. If you flip 2 coins then say whatever the first coin was the the odds if you said H was HH, or HT and if you said T it would be TH, TT. However, if you flip two coins and then say if you got at least one head independently from whatever you flipped then the odds you have 3 options HT, HH, TH with equal odds. So, the question is if the full statement was based on the data or only the truth value of the statement is based on the data. PS: Now assuming it's truth value is based on data. if you look at all options there are 14 gender day combinations per kid and 14 * 14 = 196 gender day combinations in totoal. Only 14 of of those 196 start BT which is then split evenly 7 BTB_, 7 BTG_. However that leaves 196 - 14 other options to consider. 7 * 14 of them Start G, and 6 * 14 of them start with B not on a Tuesday, but out of those you only keep 1/14 as you need BT on the second roll. Now add them up 13B and 14G out of (13 + 14) = 27. Or 13/27 B, and 14/27G.

Chinjut 9y ago

Your problem is that you are thinking there's a "the boy". But there's not a "the boy". Mr. Jones could have two boys. He could have two boys both born on Tuesday, even. The term "the boy" does not denote any particular boy, in that case, and causes you to think about the situation erroneously. If the question were "There's Kid 1 and Kid 2, each independently selected with random gender and birth-day-of-the-week. Out of those cases where Kid 1 is a boy born on Tuesday, what proportion are cases where Kid 2 is a girl?", then the answer would indeed be a straightforward 50%; the status of Kid 1 is entirely independent of the status of Kid 2. But that's not the question. The question is "There's Kid 1 and Kid 2, each independently selected with random gender and birth-day-of-the-week. Out of those cases where at least one (either one, and possibly both) of Kid 1 and Kid 2 is a boy born on Tuesday, what proportion are cases where at least one of Kid 1 and Kid 2 is a girl?". This is very different, and of course just drawing out the possibilities (all 2 * 7 * 2 * 7 equiprobable-by-stipulation choices of gender and birth-day-of-the-week for Kid 1 and Kid 2) and circling which pairs of subsets are the relevant ones for the two questions reveals the difference, the probabilities for either question elementarily calculable in this way by basic counting.

Shendare 9y ago
Agreed. He removed the second B2B2 probability annotation as though it were a repeat of the first and inapplicable to the probability set, but that's not the case, and it shouldn't be removed. Apply lower-case to the younger boy in the probability sets and it's clear why. B2b2 is not the same occurrence as b2B2. Even though the day both were born on was "a Tuesday" doesn't mean both probability instances are referring to the exact same event. Except in the case of twins, which is outside the scope of the exercise.
- deleted 9y ago
  [deleted]
Chinjut 9y ago
The 14/27 answer in the video is correct, incidentally. Also, I notice you said "Somehow knowing the day of the week the boy was born changes the result. It's completely bizarre." Remember, though, there's no "the boy". The question "On which day of the week was the boy born? Tell me, I need to know!" does not always have a well-defined answer. Indeed, you'd get the same 14/27 answer even if "Tuesday" in the question "What proportion of two-children families with at least one Tuesday boy have a girl?" was replaced by any other day. And if this seems paradoxically in conflict with the fact that simply asking "What proportion of two-children families with at least one boy have a girl?" has instead the answer 2/3, reflect again upon the fact that some families have two boys born on different days, so that there's no single answer to "On what day was 'the boy' born?". And then just draw out the cases and count. (Specifically, out of the 2 * 7 * 2 * 7 equiprobable cases overall for Kid 1 and Kid 2's genders and days, there are 27 cases where there's at least one Tuesday boy, and 14 cases where there's at least one Tuesday boy and also a girl. There are 3 * 7^2 cases where there's at least one boy, and 2 * 7^2 cases where there's at least one boy and also a girl.) Many of these questions, I think, become clearer if thought of as counting questions instead of as "probability" questions (though it's all the same; the math called "probability" is just the math of various kinds of counting (from simple counting as in this case to complexly weighted continuous measurements, but still ultimately a generalized form of counting). However, despite that equivalence, the concept "probability" has developed all these other distracting connotations, such that psychologically, there can be a useful difference in perspective in switch to explicitly thinking "counting" instead. No one would long dispute that there are 27 cases with at least one Tuesday boy, etc.).

justinpombrio 9y ago

The question is ill-posed: it does not give you enough information to tell the probability. You know what Mr. Jones has told you, but you don't know under what circumstances he would have told you this. Suppose that you ask Mr. Jones weather he has a boy and he says yes. Then the probability that he also has a girl is 2/3. Suppose that you asked Mr. Jones weather he had a boy born on a Tuesday, and he says yes. Then the probability that he has a girl is less than 2/3, because having two boys gives (about) double the chance for one of them to have been born on a Tuesday. However, suppose that you asked Mr. Jones weather he has a boy, and if so what day his eldest boy was born on, and he says "yes, and on Tuesday". Then the probability that he also has a girl is again exactly 2/3. Wikipedia has a detailed explanation: https://en.wikipedia.org/wiki/Boy_or_Girl_paradox https://en.wikipedia.org/wiki/Boy_or_Girl_paradox

Chinjut 9y ago
Everything you say after your first paragraph is correct (presuming people always answer questions with "Yes" or "No" honestly), but… No one said anything about "Mr. Jones has told you…", here. There was nothing about asking Mr. Jones a question and him providing an answer according to some process. Rather, the question was simply "Mr. Jones has two children. What is the probability he has a girl if he has a boy born on Tuesday?". There are implicit conventions involved in reading this, but not particularly problematic ones. This implicitly means "Out of all families with two children, at least one of which is a boy born on Tuesday, what proportion have a girl? [Presuming that out of those families, birth gender and day of the week for the two children are all independently uniformly distributed]". And this is a straightforward counting problem. So the wording seems fine and the problem well-posed to me.
- Retric 9y ago
  There are implicit conventions involved in reading this Explicitly the question adds no such limits. So, abstractly someone could be asking the question without those limits. It's like the difference between infinity and how whatever subset of math you work in defines infinity. And yes there are more than one commonly used definition.

astrocat 9y ago

I'm an idiot, but I'm going to throw my hat in the ring here: The video is wrong. The problem reads: Jones has 2 kids. What is P(he has a girl) given that he has a boy born on a Tuesday. Consider, for a moment, what information we're getting from "boy born on a Tuesday." This is no different than "boy with red hair," or "boy with 5 freckles." The fact that the BOY was born on a tuesday does not change P(day of the week girl was born). Imagine the "boy with 5 freckles" case - let 5 freckles be denoted by F5, six freckles by F6 and so on... would the appropriate calculation include enumerating P(boy F5, boy Fn) for all n? No. The "born on Tuesday" is irrelevant. Thus you have the following scenarios: - one kid is TuesdayBoy and the other is also a boy, born at any time - one kid is TuesdayBoy and the other is a girl, born at any time Out of these options P(Jones has a girl) is a flat out 50%. There is no need to bring in concepts of "which was born first" or enumerate all possible days of the week each child could have been born. Ok... now all the real smartypants here can correct me :)

Chinjut 9y ago
There are 2 * 7 * 2 * 7 ways to assign gender and birth-day-of-week to two children. By convention, all are considered equiprobable (this is the same as assuming kids' genders and birth day-of-weeks are independent of each other and of all facts about other kids, and that both genders are equally likely and all 7 days are equally likely for any given kid.) Of these possibilities, 27 are situations where one kid is a Tuesday boy. [Do you dispute this count?] Of those, 14 are situations where one kid is a girl. [Do you dispute this count?] The answer to "What proportion of cases where there is at least one Tuesday boy also have a girl?" is thus 14/27. You have stated by fiat that certain things are irrelevant to certain other things, that certain things have probability 50%, etc, but in doing so, you have not considered the count correctly. You are likely misled by phrasing such as "the boy", when there are families with two boys in which there is no proper referent of "the boy" and no particular answer to question like "Which day was 'the boy' born?".
- astrocat 9y ago
  ok ok... let me try to get this straight. Just as kind of a mental process for trying to understand whether or not something passes the smell test, I typically try to take the basic premise and turn it up to 11 and see if that still makes sense. In this problem, as you've described it, we're enumerating "ways to assign gender and birth-day-of-week." We can do this because there are a countable number of "days of the week" (so we can map to the integers: 1-6) AND there is also a surjective function of [child] -> [day of the week they were born]. Am I right so far? Now let's replace the set [1-6] with another countable set that also maintains the surjective function. We could say "day in the lunar cycle" (so ~27 options), or better "day of the year" (366 options), for example. Do we now need to consider the 23662366 ways to assign gender and birth-day-of-the-year? Take it further with whatever you want: "birth weight in milligrams" or "number of freckles" (as I previously suggested). All countable things that meet the surjective requirement. This is starting to smell funny, right? So let's take a look at the math. You say there are 2727 ways to configure day+gender, assuming independence for kid 1 (k1) and kid 2 (k2). This represents: (k1 gender options * k1 day of week options) * (k2 gender options * k2 day of week options). Right? I'm with you so far. Then you say "Of these possibilities, 27 are situations where one kid is a Tuesday boy." Hold up. We are given two pieces of information: that one of the kids is a boy, and that particular boy was born on a Tuesday. Let's say the boy is k1 (this is an assignment of enumeration, not of "who came first;" just like Sunday = 1 does not mean that any kid born on a Sunday was born before every kid born on Monday = 2). So now the k1 options are [11] (boy, tuesday), and the total number of options are: [11] * [27] = 14. Of those 14, 7 are girl options. And we're back to a straight 50%. So yes, I dispute the 27 number. It seems like it is arrived at by 2127, minus one for an apparent duplicate. But the 212*7 represents maintaining gender non-specificity for Tuesday boy, which should be incorrect, no? > You have stated by fiat that certain things are irrelevant to certain other things... Yes, but that's what "independent" means, right? You also stated that you're assuming these two things are independent, hence equiprobability. But independence is defined by P(A) = P(A|B). The probability of A is completely unaffected by B. Yet the outcome you arrive at is that P(A) IS affected by B, so the math presented is internally inconsistent. What am I missing here? I'm fascinated by the uncertainty around this little problem.