## 13/27

*I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?* The MetaFilter thread should eventually persuade you that the strange answer of 13/27 is correct; this comment is, I think, the clearest.

the smartest kid on earth

*I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?* The MetaFilter thread should eventually persuade you that the strange answer of 13/27 is correct; this comment is, I think, the clearest.

Written by gerrycanavan

May 26, 2010 at 12:59 am

Posted in Look at what I found on the Internet

Tagged with math, our brains don't work, probability, weird science

Subscribe to comments with RSS.

- Photo: he’s right behind me, isn’t he tmblr.co/Z-JE0w1MCJo1u 2 hours ago
- @jhrees @pankisseskafka liars, murderers, thieves 2 hours ago
- RT @Exhaust_Fumes: I will not get riled upI will not get riled upI will not get riled upI will not get riled upI will not get riled upI wil… 2 hours ago
- You're a monster, why can't you see it RT @jhrees: @pankisseskafka You lost me there. You have to enter administration to become management. 2 hours ago
- If TT faculty know everything, they must know we suffer; if TT faculty can do anything, they could end our suffering; therefore... 2 hours ago
- of course, this is already forever true on the right RT @gerrycanavan: BREAKING: Emeritus professor yells at cloud news.wisc.edu/23011 3 hours ago

- mattintoledo on Tuesday Links!
- mattintoledo on Wednesday Links!
- mattintoledo on Monday Links!
- mattintoledo on Sunday Links for the Sunday Reader
- A. Reader on Happy Birthday Connor Links!
- Infinite Tasks on Happy Birthday Connor Links!

- Tuesday Links!
- Sunday Links!
- Friday Links!
- Thursday Links!
- Tuesday Morning Links
- Special! Bonus! Weekend! Links!

- Tuesday Links!
- Why You're So Fat
- Tuesday Morning Links
- Friday Links!
- Grown-Up Calvin and Hobbes
- Sunday Links!
- And That's Before Getting Into the Whole 'Roko's Basilisk' Mess
- Infinite Summer #5: Maps and Territories
- Syllabus: 'Science Fiction and Ecology'
- Monkey Torture

- 3 Quarks Daily
- A&L Daily
- Acephalous
- Ads without Products
- Ain't It Cool News
- Amazon.com
- An und für sich
- Ballardian
- Barking Up the Wrong Tree
- Bitter Laughter
- Blastr
- Bleeding Cool
- Boing Boing
- Bookninja
- Bookslut
- Candleblog
- Chronicle of Higher Education
- Climate Progress
- Comedy Bang Bang
- Crooked Timber
- Crooks and Liars
- Cura Te Ipsum
- Cynical-C
- Dan Hassler-Forest
- Dial B for Blog
- Disaster Notes
- Dot Earth
- Ecology without Nature
- Endangered Durham
- Ezra Klein
- Feministe
- Fraggmented
- Freddie deBoer
- Glenn Greenwald
- Gravity Lens
- Grist
- Hero Complex
- How the University Works
- Hullabaloo
- In Focus
- Independent Weekly
- Inside Higher Ed
- Inside Higher Ed
- io9
- Jason Kottke
- Jay Is Games
- Lawyers, Guns, and Money
- Lenin's Tomb
- Let's Get Critical
- Letters of Note
- Long Reads
- Longform
- Marquette University English Department
- Marquette University English Department Blog
- Matt Taibbi
- Matt Yglesias
- McSweeney's
- MetaFilter
- Michael Bérubé
- Mightygodking
- Monthly Review
- MS Paint Adventures
- Nate Silver
- Neilalien
- Nerdist
- New Left Review
- Pandagon
- Paul F. Tompkins
- Philosophy of Water
- Poli-Sci-Fi Radio
- Polygraph
- rejectamentalist manifesto
- Reviews in Cultural Theory
- SF Signal
- Socialism and/or Barbarism
- Steve Benen
- Strange Maps
- Student Activism
- Ta-Nehisi Coates
- Talking Points Memo
- Television without Pity
- The Big Picture
- The House Next Door
- The Incomparable
- The Pinocchio Theory
- The Rushmore Academy
- Think Progress
- This American Life
- Tom Tomorrow
- Toothpaste for Dinner
- TV Tropes
- Washington Monthly
- Waxy
- Whedonesque
- WTF with Marc Maron
- xkcd
- Ze Frank
- zunguzungu

academia
academic jobs
actually existing media bias
adjuncts
America
apocalypse
art
austerity
Barack Obama
books
Bush
California
capitalism
class struggle
climate change
comics
Democrats
Duke
ecology
energy
feminism
film
Fox News
futurity
games
gay rights
general election 2008
general election 2012
guns
health care
Hillary Clinton
How the University Works
John McCain
Joss Whedon
kids today
labor
literature
marriage equality
misogyny
Mitt Romney
MOOCs
music
neoliberalism
New Jersey
New York
North Carolina
nuclearity
oil
outer space
over-educated literary theory PhDs
pedagogy
politics
polls
race
rape culture
religion
Republicans
Sarah Palin
science
science fiction
student debt
Supreme Court
taxes
television
the courts
the economy
the law
the Senate
unions
war on education
what it is I think I'm doing
Wisconsin
Won't somebody think of the children?
writing
YouTube

%d bloggers like this:

Why are they assuming that the order of the children matters at all? If one of my children – and the placement doesn’t matter – is a boy born on a Tuesday, then the odds are 50/50, as long as we don’t confuse ourselves by pretending that the order matters, i.e., that it makes a difference whether child 1 is born on a Tuesday or child 2 is born on a Tuesday. Unfortunately, I’m not enough of a statistician to confront this, but it seems to me that there’s got to be a fallacy out there somewhere, with a name, that refers to precisely this kind of thing.

AlexMay 26, 2010 at 10:29 am

To make it more clear: if I go to the example that many in the mefi thread were using of generic boy, where “one of them is a boy”, and where you get BB, BG, or GB, and then I say, “BG and GB are the same,” then the odds become 50%, which is what they are IMO.

AlexMay 26, 2010 at 10:41 am

Well, having stared at this until I understood is last night, I see where you’re hung up because it’s where I was hung up last night.

The first thing to say is that birth order is a red herring; it’s not meaningful in itself but just as a way to differentiate the two children from one another. It could just as easily be “taller/shorter” or “stands on the left at family pictures”; we just need a way to identify them so they’re unique.

To take the first problem (“I have two children and one of them is a boy. What is the probability that the other is also a boy?”):

If I have two children, they can be:

BG BB GB GG

Each of these cases has a 25% chance of being true. You might object here and say GB and BG describe the same state of affairs because birth order doesn’t matter, and that’s true — but at the same time the one-each case is twice as likely as BB and GG because there’s two ways it can happen.

So if I have two children there’s a 25% chance that they’re both boys, a 50% chance that they’re one-each, and a 25% chance that they’re both girls. But the question specifies that GG didn’t happen. So we scratch that and are left with 25%/75% that they’re both boys and 50%/75% that they’re one-each. This is to say, if you check all the couples in the world that have one boy, you’ll find that 2/3 of the time the other child is a girl and 1/3 of the time the other child is a boy.

gerrycanavanMay 26, 2010 at 10:54 am

The second problem is harder to believe but it follows the same logic. There are 28 ways that you can have a boy born on a Tuesday. If the first child is the boy born on a Tuesday, then the second child is either a girl born on M,T,W,R,F,Sat,Sun (7 cases, half the time) or a boy born on M,T,W,R,F,Sat,Sun (7 cases, half the time). If the second child is the boy born on a Tuesday, you have the same 14 cases, half with the other child being a boy and half with the other child being a girl.

But one of those cases was double-counted: (BTu, BTu). So you have to eliminate it, which leaves you with 13/27. Really close to 1/2, but not quite half yet.

gerrycanavanMay 26, 2010 at 10:58 am

The way this comment explains it is helpful: the more uniquely you can identify the one boy, the closer the math will get to 1/2. For “I have two children, one of them is President Barack Obama. What is the gender of the other?” the odds are 50-50.

gerrycanavanMay 26, 2010 at 11:01 am

Here’s the thing about the BG BB GB GG thing: it’s a complete delusion. Let me take a more concrete example: I have a boy in my arms. I tell someone, “This is my son. What are the odds that my other child is a boy.” It makes no difference whatsoever, whether the child is older or younger, and therefore, GB and BG are exactly the same thing – you can’t distinguish them, because they’re not different for the purposes of our problem. So the odds of the “other child,” older or younger being of no material import, being a boy is 50%. You say that birth order is a red herring, but in your calculus you’re including birth order as something that matters. I’m saying birth order is a red herring, so I’m getting rid of it. The same goes for the Tuesday example: “This is my son; he was born on a Tuesday. What are the odds of my other child being a boy.” Unlike in the (imaginary) example in which birth order might matter, Tuesday is now of no import whatsoever, and therefore, the odds are 50% because BG and GB are the same thing.

AlexMay 26, 2010 at 11:20 am

I perfectly understand the logic of the 13/27, and I’m saying it’s wrong.

AlexMay 26, 2010 at 11:24 am

Oh, and GG is irrelevant because we already know that one of the children is a boy.

AlexMay 26, 2010 at 11:26 am

The version you’re using — this is my son — is the Barack Obama version. You’ve revealed the gender of one of the specific children, so the odds are fifty-fifty for the other.

In the original version of the problem it’s not “this is a boy” but “one of the two is a boy.” That tiny logical distinction is what shifts the odds.

If you think of it with coin flips it might be easier. Flip two coins and hide the results from me. The odds are:

HH 25%

HT/TH 50%

TT 25%

If you tell me that one of the coins is H, then I know that 1/3 of the time they’ll both be heads and 2/3 of the time the other will be tails. But if you show me Coin #1 came up heads, then the odds are 50-50 for the other.

gerrycanavanMay 26, 2010 at 11:30 am

Here’s what “Nothing” says on the thread:

This seems like an annoying trick, not a puzzle. The difference is between the cases where order is important and where it is not. By specifying a distinguishing characteristic for the boy, you are forcing people to consider order and BB becomes bB and Bb.

But really nothing has changed. GB and GB are not equally probable to BB or GG. They are exactly half as probable.

AlexMay 26, 2010 at 11:30 am

This is what I was saying earlier about how birth order doesn’t matter; it’s just the most convenient way to name one child #1 and the other #2.

But you have to order the children because they’re separate events. Otherwise we’d have to conclude that BB, GB/BG, and GG are all equally likely, when we know they’re not. GB/BG is twice as likely as the others.

gerrycanavanMay 26, 2010 at 11:34 am

Nothing has it wrong. Think of the coin example. You have to “order” separate events in some way because they have separate probabilities you can’t conflate. When you say “one of my two children is a boy” that’s probabilistically different from saying “this child (the oldest, the tallest, the one standing in front of me) is a boy.”

gerrycanavanMay 26, 2010 at 11:40 am

Here’s the problem with the coin example: you’re assuming that the flipper is flipping both coins, and then saying, “One of them is a head.” But in the boy example, that’s not happening. He’s already flipped a coin, got a head, and said, “One of them is a boy.” You just have to identify what the second one is.

AlexMay 26, 2010 at 11:51 am

In other words, imagine that the coin flipper has a head in his hand. He puts it behind his back. Then he flips another coin, puts that behind his back. He scrambles up the order, but we know that one of them was a head. He says: “One of them is a head,” takes a head out from behind his back. What are the odds of the other being a head?

The coin flip example as you put it is misleading.

AlexMay 26, 2010 at 11:54 am

Here’s the problem with the coin example: you’re assuming that the flipper is flipping both coins, and then saying, “One of them is a head.” But in the boy example, that’s not happening. He’s already flipped a coin, got a head, and said, “One of them is a boy.” You just have to identify what the second one is.That’s actually what you’re doing. In the original question, the person has had two children (flipped two coins) and told you one was a boy (heads). In your version they’re flipping one coin, having it come up heads, and then flipping the second. You’ve assumed a particular order for the children when you don’t have that information; that’s why the odds that the second child is a boy are 1/3 and not 1/2.

gerrycanavanMay 26, 2010 at 11:55 am

In other words, imagine that the coin flipper has a head in his hand. He puts it behind his back. Then he flips another coin, puts that behind his back. He scrambles up the order, but we know that one of them was a head. He says: “One of them is a head,” takes a head out from behind his back. What are the odds of the other being a head?This makes the same error. You’re saying the FIRST child is a boy. The question only says that one of the two is.gerrycanavanMay 26, 2010 at 11:58 am

You’re right. I thought I had more arguments, but i just constructed the following imaginary scenario: 100 families have two children each. The odds of each child being a boy or girl is 50%. Pick one of these families that has a boy. What are the odds that the other is a boy? 1/3. The rest of the problem follows from there.

AlexMay 26, 2010 at 12:33 pm

Let’s never fight again.

gerrycanavanMay 26, 2010 at 12:36 pm

I’m going to be honest with everyone. This thing absolutely ruined my Tuesday night. I was in a hotel room in Mississippi until 3 am trying to get to the bottom of it. I only really got as far as understanding the 1/3 probability of the first question.

Shankar DMay 27, 2010 at 5:53 pm