5 ms·

It's bedtime here so I'm just going to chuck this in and run away: when you've got Monty Hall sussed, try Tuesday's Child, i.e. "I have two children, one of the

by hanoz 7y ago

It's bedtime here so I'm just going to chuck this in and run away: when you've got Monty Hall sussed, try Tuesday's Child, i.e. "I have two children, one of them is a boy born on a Tuesday, what is the probability I have two boys?"

earenndil 7y ago
Is it not 50%?
- hanoz 7y ago
  It is not.
  - deleted 7y ago
    [deleted]
  - deleted 7y ago
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- nwallin 7y ago
  Having once male child eliminates the possibly of having two female children. You take a four entry truth table and scratch one out. The truth table is male->female, female->male, male->male. Tuesday is a red herring. Note that there's some ambiguity between exactly one of vs at least one of which changes the probability but not as much as 1/2->1/3. (exactly one male on a Tuesday excludes the possibility of two males born on a Tuesday) Tuesday is not a red herring. It's truth tables all the way down.
  - AnotherGoodName 7y ago
    Ahh i see. It's because the ordering doesn't matter. If the question made the ordering matter; "I have two children, the youngest is a boy, what is the probability that i have two boys" Then it's definitely 50%. The truth table would be Youngest: Boy Oldest: Girl Youngest: Boy Oldest: Boy Youngest: Girl Oldest: Girl Youngest: Girl Oldest: Boy With the bottom two being ruled out by the question. I think that's the part that catches people out here.
fullerite 7y ago
1/3
chowells 7y ago
You're going to say it's 1/3. I'm going to say that the problem is not described clearly enough to assert that. For instance, let's say you tell me you have two children. One was born on a Tuesday, the other on a Wednesday. I ask about the one born on Tuesday, and you tell me it's a boy. With the information I have available to me at this point, I am correct to judge that the odds you have two boys are 1/2. This story differs from the one you provided in ways that are insignificant for story telling. The way you tell the story depends on the reader picking up on a rather tiny detail to reach the conclusion you intend. Specifically, the reader must come to the conclusion that you are someone playing games with them who has intentionally used very careful wording to only eliminate the girl/girl entry from the outcome chart. That's not the way people communicate naturally. If someone says "I have two kids. One's a boy...", they aren't playing probability games, they're thinking of a child they'd like to tell you a story about. In that case, when they have a distinguished child in mind before making that statement, the odds are in fact 1/2 that the other child is a boy and hence you have two boys. (Modulo all the legitimate criticisms that can be raised that there are more possible outcomes than boy/girl and that even between those two the odds aren't equal. This is a simplification of reality for the sake of a puzzle.) In the end, problems like this mostly hinge on poor communication that abuses common patterns to convey less information than the reader interprets from their experiences communicating in the real world. As such, they're really rather boring.
- hanoz 7y ago
  > You're going to say it's 1/3. I most certainly am not. I'm going to say it's 13/27. I agree the imprecise nature of the English language muddies the water somewhat, but the question can be re-framed to remove this aspect and yet still have people give the wrong answer, and as such is not just a "really rather boring" case of "intentional" "poor communication", but offers an interesting insight into the shortcomings of our intuition for probability. So let's set aside the original problem and put this one instead (which may or may not be equivalent to the original depending on your interpretation, but let's take this in isolation anyway): Woman: How may children do you have? Man: Two. Woman: Is at least one of them a boy who was born on a Tuesday? Man: Yes. What now is the probability the man has two boys? The answer is in fact 13/27, because of the unlikely extra information revealed by the Tuesday condition, which I think is surprising and interesting.
eximius 7y ago
This doesn't seem the same because we don't pick something before you give us all of the information, no?
Sean1708 7y ago
Am I being really stupid here? I keep getting 13/27, both by Baye's Theory P(BTB?∩B?BT | BT??∩??BT) = P(BT??∩??BT | BTB?∩B?BT) · P(BTB?∩B?BT) / P(BT??∩??BT) = 1 · (1/28 + 1/28 - 1/196) / (1/14 + 1/14 - 1/196) = 13/27 and by listing out all the combinations Boy Tuesday │ Boy Tuesday Boy Any ────────────┼──────────────────── B Mo B Tu │B Mo B Tu B Tu B Mo │B Tu B Mo B Tu B Tu │B Tu B Tu B Tu B We │B Tu B We B Tu B Th │B Tu B Th B Tu B Fr │B Tu B Fr B Tu B Sa │B Tu B Sa B Tu B Su │B Tu B Su B Tu G Mo │ B Tu G Tu │ B Tu G We │ B Tu G Th │ B Tu G Fr │ B Tu G Sa │ B Tu G Su │ B We B Tu │B We B Tu B Th B Tu │B Th B Tu B Fr B Tu │B Fr B Tu B Sa B Tu │B Sa B Tu B Su B Tu │B Su B Tu G Mo B Tu │ G Tu B Tu │ G We B Tu │ G Th B Tu │ G Fr B Tu │ G Sa B Tu │ G Su B Tu │
- hanoz 7y ago
  You are correct! Most people tend to say 1/2, or if they have some knowledge of probability but dismiss the Tuesday aspect then 1/3. It does rely on the problem being interpreted as equivalent to the dialogue I have posted below, but even when so framed, many people will refuse to believe it.
- username90 7y ago
  You are not correct, but you are not entirely wrong. This is a badly made retake of the original two problems where you had one where you just mention that you have a boy, and the other where you mention that the older is a boy. The older being a boy removes 2 cases while 1 being a boy just removes 1 case so they are different. But in this case mentioning the weekday doesn't matter at all. To make it clearer take this script "I have 2 kids, one is a boy, that boy's birthday is {{ boy's birthday }}". In that case it is 1/3. Take this script on the other hand "I have 2 kids {{ if 1 is a boy born on tuesday print ', one which is a boy born on Tuesday' }}". That is what you and he meant it. But it is wrong since it is not the only way to read it, and arguably it is the less correct way to read it. This HN thread is full of people who don't know maths who think they know it and are trying to teach others...
  - hanoz 7y ago

by hanoz 7y ago

earenndil 7y ago

Is it not 50%?

hanoz 7y ago
It is not.
- deleted 7y ago
  [deleted]
- deleted 7y ago
  [deleted]
nwallin 7y ago
Having once male child eliminates the possibly of having two female children. You take a four entry truth table and scratch one out. The truth table is male->female, female->male, male->male. Tuesday is a red herring. Note that there's some ambiguity between exactly one of vs at least one of which changes the probability but not as much as 1/2->1/3. (exactly one male on a Tuesday excludes the possibility of two males born on a Tuesday) Tuesday is not a red herring. It's truth tables all the way down.
- AnotherGoodName 7y ago
  Ahh i see. It's because the ordering doesn't matter. If the question made the ordering matter; "I have two children, the youngest is a boy, what is the probability that i have two boys" Then it's definitely 50%. The truth table would be Youngest: Boy Oldest: Girl Youngest: Boy Oldest: Boy Youngest: Girl Oldest: Girl Youngest: Girl Oldest: Boy With the bottom two being ruled out by the question. I think that's the part that catches people out here.

fullerite 7y ago

1/3

chowells 7y ago

You're going to say it's 1/3. I'm going to say that the problem is not described clearly enough to assert that. For instance, let's say you tell me you have two children. One was born on a Tuesday, the other on a Wednesday. I ask about the one born on Tuesday, and you tell me it's a boy. With the information I have available to me at this point, I am correct to judge that the odds you have two boys are 1/2. This story differs from the one you provided in ways that are insignificant for story telling. The way you tell the story depends on the reader picking up on a rather tiny detail to reach the conclusion you intend. Specifically, the reader must come to the conclusion that you are someone playing games with them who has intentionally used very careful wording to only eliminate the girl/girl entry from the outcome chart. That's not the way people communicate naturally. If someone says "I have two kids. One's a boy...", they aren't playing probability games, they're thinking of a child they'd like to tell you a story about. In that case, when they have a distinguished child in mind before making that statement, the odds are in fact 1/2 that the other child is a boy and hence you have two boys. (Modulo all the legitimate criticisms that can be raised that there are more possible outcomes than boy/girl and that even between those two the odds aren't equal. This is a simplification of reality for the sake of a puzzle.) In the end, problems like this mostly hinge on poor communication that abuses common patterns to convey less information than the reader interprets from their experiences communicating in the real world. As such, they're really rather boring.

hanoz 7y ago
> You're going to say it's 1/3. I most certainly am not. I'm going to say it's 13/27. I agree the imprecise nature of the English language muddies the water somewhat, but the question can be re-framed to remove this aspect and yet still have people give the wrong answer, and as such is not just a "really rather boring" case of "intentional" "poor communication", but offers an interesting insight into the shortcomings of our intuition for probability. So let's set aside the original problem and put this one instead (which may or may not be equivalent to the original depending on your interpretation, but let's take this in isolation anyway): Woman: How may children do you have? Man: Two. Woman: Is at least one of them a boy who was born on a Tuesday? Man: Yes. What now is the probability the man has two boys? The answer is in fact 13/27, because of the unlikely extra information revealed by the Tuesday condition, which I think is surprising and interesting.

eximius 7y ago

This doesn't seem the same because we don't pick something before you give us all of the information, no?

Sean1708 7y ago

Am I being really stupid here? I keep getting 13/27, both by Baye's Theory P(BTB?∩B?BT | BT??∩??BT) = P(BT??∩??BT | BTB?∩B?BT) · P(BTB?∩B?BT) / P(BT??∩??BT) = 1 · (1/28 + 1/28 - 1/196) / (1/14 + 1/14 - 1/196) = 13/27 and by listing out all the combinations Boy Tuesday │ Boy Tuesday Boy Any ────────────┼──────────────────── B Mo B Tu │B Mo B Tu B Tu B Mo │B Tu B Mo B Tu B Tu │B Tu B Tu B Tu B We │B Tu B We B Tu B Th │B Tu B Th B Tu B Fr │B Tu B Fr B Tu B Sa │B Tu B Sa B Tu B Su │B Tu B Su B Tu G Mo │ B Tu G Tu │ B Tu G We │ B Tu G Th │ B Tu G Fr │ B Tu G Sa │ B Tu G Su │ B We B Tu │B We B Tu B Th B Tu │B Th B Tu B Fr B Tu │B Fr B Tu B Sa B Tu │B Sa B Tu B Su B Tu │B Su B Tu G Mo B Tu │ G Tu B Tu │ G We B Tu │ G Th B Tu │ G Fr B Tu │ G Sa B Tu │ G Su B Tu │

hanoz 7y ago
You are correct! Most people tend to say 1/2, or if they have some knowledge of probability but dismiss the Tuesday aspect then 1/3. It does rely on the problem being interpreted as equivalent to the dialogue I have posted below, but even when so framed, many people will refuse to believe it.
username90 7y ago
You are not correct, but you are not entirely wrong. This is a badly made retake of the original two problems where you had one where you just mention that you have a boy, and the other where you mention that the older is a boy. The older being a boy removes 2 cases while 1 being a boy just removes 1 case so they are different. But in this case mentioning the weekday doesn't matter at all. To make it clearer take this script "I have 2 kids, one is a boy, that boy's birthday is {{ boy's birthday }}". In that case it is 1/3. Take this script on the other hand "I have 2 kids {{ if 1 is a boy born on tuesday print ', one which is a boy born on Tuesday' }}". That is what you and he meant it. But it is wrong since it is not the only way to read it, and arguably it is the less correct way to read it. This HN thread is full of people who don't know maths who think they know it and are trying to teach others...
- hanoz 7y ago