From 7ded418511238e790b43b90b8225db04d1d2a087 Mon Sep 17 00:00:00 2001 From: Kyle Shankwiler Date: Sat, 16 Nov 2024 12:12:27 -0800 Subject: [PATCH] boy girl paradox post --- .gitignore | 1 + posts/boy-girl-paradox/back-office.png | Bin 0 -> 47193 bytes .../boy-girl-paradox/earnest-wrong-answer.png | Bin 0 -> 60583 bytes posts/boy-girl-paradox/index.html | 297 ++++++++++++++++++ posts/boy-girl-paradox/instagram-post.png | Bin 0 -> 109187 bytes posts/boy-girl-paradox/reevaluate.png | Bin 0 -> 40875 bytes posts/boy-girl-paradox/right-answer.png | Bin 0 -> 32499 bytes posts/boy-girl-paradox/style.css | 35 +++ posts/boy-girl-paradox/walked-it-back.png | Bin 0 -> 25430 bytes posts/style.css | 49 +++ 10 files changed, 382 insertions(+) create mode 100644 .gitignore create mode 100644 posts/boy-girl-paradox/back-office.png create mode 100644 posts/boy-girl-paradox/earnest-wrong-answer.png create mode 100644 posts/boy-girl-paradox/index.html create mode 100644 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+

+ You call a family that has two children. A boy picks up. What is the + probability that the other is a boy? +

+

The Prompt

+
+
+ +
+ Instagram post +
+
+
+

Lots of Disagreement

+
+
+
+ +
reevaluate it
+
+
+
+
+ +
reasonable but not quite right
+
+
+
+
+ +
back office
+
+
+
+
+ +
nice
+
+
+
+
+ +
didn't stick to his guns
+
+
+
+

+ The most liked comment answers 1/2, but the user got worn down by others + until he recanted and agreed that the answer is 1/3. Many other comments + give lengthy explanations as to why the answer is 1/3. +

+

The actual answer is 1/2.

+

Framing the Question

+

+ Let's flesh out this hypothetical, to make it easier to reason about. +

+
    +
  • + You're a salesman who sells bunkbeds, specifically bunkbeds that only + work for two boys. +
  • +
  • + You have a phonebook of leads, containing the phone number of all + households in the country that have exactly two kids. +
  • +
  • + When you make a call, you anxiously wait to find out whether you have + called the right kind of home. If the home has two girls or a girl and + a boy, you've wasted your time. +
  • +
  • + When you call, there's an equal probability that either child answers. +
  • +
  • + When a boy answers, you're happy. The household has at least one boy. +
  • +
  • + But how happy should you be? How likely is it, given that the kid + answering the phone is a boy, that the other kid is also a boy? +
  • +
+ +

Explanation

+

+ First, imagine there are 24 households with kids. The distribution of + genders in those households is as expected, with a quarter of them (6) + having two girls, half of them (12) having a girl and boy, and a quarter + of them (6) having two boys. +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
GGGGGGGGGGGG
BGBGBGBGBGBG
BGBGBGBGBGBG
BBBBBBBBBBBB
+

+ An intuitive explanation is: look at all the Bs on the board. How many + have a brother? Half. A longer explanation is as follows. +

+

We want to solve P(called row 4 given that a boy picked up).

+
    +
  • P(called row 4) = 1/4
  • +
  • + P(a boy picked up) = 1/2, because: +
      +
    • + half the children are boys, each family is equally likely to have + been called, and each child in the family is equally likely to + pick up. +
    • +
    +
  • +
  • + To re-emphasize, P(a boy picked up) = +
      +
    • + P(a boy picks up from row 1) + P(a boy picks up from row 2) + P(a + boy picks up from row 3) + P(a boy picks up from row 4) = +
    • +
    • + ((1/4) * 0) + ((1/4) * (1/2)) + ((1/4) * (1/2)) + ((1/4) * 1) = +
    • +
    • 1/2.
    • +
    +
  • +
  • + Therefore, P(called row 4 given that a boy picked up) = +
      +
    • + (P(a boy picked up given that row 4 was called) * P(row 4 was + called)) / P(a boy was called) = +
    • +
    • (1 * (1/4)) / (1/2) =
    • +
    • 1/2.
    • +
    • + The link between the first bullet in this nested list and the + second is given by Baye's Theorem. +
    • +
    +
  • +
+ +

The answer is 1/2.

+ +

Why do so many people think the answer is 1/3?

+ +

+ The line of reasoning getting people to 1/3 is something like the + following: +

+
    +
  • + After calling and discovering that at least one boy is in the family, + row 1 is eliminated. +
  • +
  • Of the 18 remaining families, 6 have two boys.
  • +
  • + Therefore, you have a 6/18 = 1/3 chance of having called a home with + two boys. +
  • +
+ +

+ The flaw there is that you're ignoring the fact that for row 4, there's + a 100% chance a boy answers the phone, whereas for rows 2 and 3, there's + a 50% chance a boy answers the phone. +

+ +

Explicitly, P(called row 2 column 1 given that a boy answered) =

+
    +
  • + (P(a boy answered given row 2 column 1 was called) * P(row 2 column 1 + was called)) / P(a boy was called) = +
  • +
  • ((1/2) * (1/24)) / (1/2) =
  • +
  • 1/24
  • +
+ +

+ Similar logic can be used to conclude P(called row 4 column 1 given that + a boy answered) = 1/12. +

+ +

+ You know as soon as the boy answers that you're twice as likely to be + talking to row 4 column 1 than row 2 column 1. But there are twice as + many BG families than BB families, so that evens out, and you have an + equal probability of talking to a BG family as a BB family, once you + know a boy answered. +

+ +

In what situation would the answer be 1/3?

+ +

+ If the question was phrased as "You are calling a family with 2 children + with at least one boy. What's the probability that the family has 2 + boys?" then 1/3 would be the answer. +

+ +

+ You could imagine that our salesman, instead of having a list of all + families with 2 kids, has a list of all families with 2 kids with at + least one boy. When he picks up the phone, it's ringing, and he's + anxiously waiting to hear the voice on the other line. He knows it's one + of the families from rows 2 to 4, and it's equally likely to be each of + them. Therefore, there's a 6 / 18 chance that it's row 4. +

+ +

+ The key there is that there was no event, a boy picking up, which + indicated a weighting toward one family versus another. All families in + the set are equally likely from the salesman's perspective. +

+ +

Conclusion

+ +

+ I think it's pretty funny that people got so dug in on the 1/3 thing. It + feels like they were so primed for expecting a trick question that it + felt like it had to be the unexpected 1/3 rather than the obvious 1/2. +

+ +

+ The subtleties and confusion around this question are captured by + Wikipedia's + Boy or Girl Paradox. Looking at the "Second Question" section, you can see that our + question is equivalent to one of their examples: "From all families with + two children, one child is selected at random, and the sex of that child + is specified to be a boy. This would yield an answer of 1/2." +

+
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