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.DS_Store |
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<!DOCTYPE html> | ||
<!-- Don't mind the markup and CSS mess, this is just for fun.--> | ||
<html> | ||
<head> | ||
<meta charset="UTF-8" /> | ||
<meta name="viewport" content="width=device-width, initial-scale=1" /> | ||
<link | ||
href="https://fonts.googleapis.com/css?family=Roboto" | ||
rel="stylesheet" | ||
/> | ||
<link href="../style.css" rel="stylesheet" /> | ||
<link href="style.css" rel="stylesheet" /> | ||
</head> | ||
<body> | ||
<div class="content"> | ||
<h1> | ||
You call a family that has two children. A boy picks up. What is the | ||
probability that the other is a boy? | ||
</h1> | ||
<h2>The Prompt</h2> | ||
<div | ||
style=" | ||
width: 100%; | ||
display: flex; | ||
align-items: center; | ||
justify-content: center; | ||
" | ||
> | ||
<figure> | ||
<img src="instagram-post.png" width="300" /> | ||
<figcaption> | ||
<a | ||
href="https://www.instagram.com/p/DCUGwCJPgyw/?utm_source=ig_web_copy_link" | ||
>Instagram post</a | ||
> | ||
</figcaption> | ||
</figure> | ||
</div> | ||
<h2>Lots of Disagreement</h2> | ||
<div | ||
style=" | ||
display: flex; | ||
width: 100%; | ||
align-content: space-between; | ||
justify-content: space-evenly; | ||
flex-wrap: wrap; | ||
align-items: center; | ||
" | ||
> | ||
<div class="comment"> | ||
<figure> | ||
<img src="reevaluate.png" width="300" /> | ||
<figcaption>reevaluate it</figcaption> | ||
</figure> | ||
</div> | ||
<div class="comment"> | ||
<figure> | ||
<img src="earnest-wrong-answer.png" width="300" /> | ||
<figcaption>reasonable but not quite right</figcaption> | ||
</figure> | ||
</div> | ||
<div class="comment"> | ||
<figure> | ||
<img src="back-office.png" width="300" /> | ||
<figcaption>back office</figcaption> | ||
</figure> | ||
</div> | ||
<div class="comment"> | ||
<figure> | ||
<img src="right-answer.png" width="300" /> | ||
<figcaption>nice</figcaption> | ||
</figure> | ||
</div> | ||
<div class="comment"> | ||
<figure> | ||
<img src="walked-it-back.png" width="300" /> | ||
<figcaption>didn't stick to his guns</figcaption> | ||
</figure> | ||
</div> | ||
</div> | ||
<p> | ||
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. | ||
</p> | ||
<p>The actual answer is 1/2.</p> | ||
<h2>Framing the Question</h2> | ||
<p> | ||
Let's flesh out this hypothetical, to make it easier to reason about. | ||
</p> | ||
<ul> | ||
<li> | ||
You're a salesman who sells bunkbeds, specifically bunkbeds that only | ||
work for two boys. | ||
</li> | ||
<li> | ||
You have a phonebook of leads, containing the phone number of all | ||
households in the country that have exactly two kids. | ||
</li> | ||
<li> | ||
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. | ||
</li> | ||
<li> | ||
When you call, there's an equal probability that either child answers. | ||
</li> | ||
<li> | ||
When a boy answers, you're happy. The household has at least one boy. | ||
</li> | ||
<li> | ||
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? | ||
</li> | ||
</ul> | ||
|
||
<h2>Explanation</h2> | ||
<p> | ||
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. | ||
</p> | ||
<table> | ||
<tr> | ||
<td class="gg">GG</td> | ||
<td class="gg">GG</td> | ||
<td class="gg">GG</td> | ||
<td class="gg">GG</td> | ||
<td class="gg">GG</td> | ||
<td class="gg">GG</td> | ||
</tr> | ||
<tr> | ||
<td class="bg">BG</td> | ||
<td class="bg">BG</td> | ||
<td class="bg">BG</td> | ||
<td class="bg">BG</td> | ||
<td class="bg">BG</td> | ||
<td class="bg">BG</td> | ||
</tr> | ||
<tr> | ||
<td class="bg">BG</td> | ||
<td class="bg">BG</td> | ||
<td class="bg">BG</td> | ||
<td class="bg">BG</td> | ||
<td class="bg">BG</td> | ||
<td class="bg">BG</td> | ||
</tr> | ||
<tr> | ||
<td class="bb">BB</td> | ||
<td class="bb">BB</td> | ||
<td class="bb">BB</td> | ||
<td class="bb">BB</td> | ||
<td class="bb">BB</td> | ||
<td class="bb">BB</td> | ||
</tr> | ||
</table> | ||
<p> | ||
An intuitive explanation is: look at all the Bs on the board. How many | ||
have a brother? Half. A longer explanation is as follows. | ||
</p> | ||
<p>We want to solve P(called row 4 given that a boy picked up).</p> | ||
<ul> | ||
<li>P(called row 4) = 1/4</li> | ||
<li> | ||
P(a boy picked up) = 1/2, because: | ||
<ul> | ||
<li> | ||
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. | ||
</li> | ||
</ul> | ||
</li> | ||
<li> | ||
To re-emphasize, P(a boy picked up) = | ||
<ul> | ||
<li> | ||
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) = | ||
</li> | ||
<li> | ||
((1/4) * 0) + ((1/4) * (1/2)) + ((1/4) * (1/2)) + ((1/4) * 1) = | ||
</li> | ||
<li>1/2.</li> | ||
</ul> | ||
</li> | ||
<li> | ||
Therefore, P(called row 4 given that a boy picked up) = | ||
<ul> | ||
<li> | ||
(P(a boy picked up given that row 4 was called) * P(row 4 was | ||
called)) / P(a boy was called) = | ||
</li> | ||
<li>(1 * (1/4)) / (1/2) =</li> | ||
<li>1/2.</li> | ||
<li> | ||
The link between the first bullet in this nested list and the | ||
second is given by Baye's Theorem. | ||
</li> | ||
</ul> | ||
</li> | ||
</ul> | ||
|
||
<p>The answer is 1/2.</p> | ||
|
||
<h2>Why do so many people think the answer is 1/3?</h2> | ||
|
||
<p> | ||
The line of reasoning getting people to 1/3 is something like the | ||
following: | ||
</p> | ||
<ul> | ||
<li> | ||
After calling and discovering that at least one boy is in the family, | ||
row 1 is eliminated. | ||
</li> | ||
<li>Of the 18 remaining families, 6 have two boys.</li> | ||
<li> | ||
Therefore, you have a 6/18 = 1/3 chance of having called a home with | ||
two boys. | ||
</li> | ||
</ul> | ||
|
||
<p> | ||
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. | ||
</p> | ||
|
||
<p>Explicitly, P(called row 2 column 1 given that a boy answered) =</p> | ||
<ul> | ||
<li> | ||
(P(a boy answered given row 2 column 1 was called) * P(row 2 column 1 | ||
was called)) / P(a boy was called) = | ||
</li> | ||
<li>((1/2) * (1/24)) / (1/2) =</li> | ||
<li>1/24</li> | ||
</ul> | ||
|
||
<p> | ||
Similar logic can be used to conclude P(called row 4 column 1 given that | ||
a boy answered) = 1/12. | ||
</p> | ||
|
||
<p> | ||
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. | ||
</p> | ||
|
||
<h2>In what situation would the answer be 1/3?</h2> | ||
|
||
<p> | ||
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. | ||
</p> | ||
|
||
<p> | ||
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. | ||
</p> | ||
|
||
<p> | ||
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. | ||
</p> | ||
|
||
<h2>Conclusion</h2> | ||
|
||
<p> | ||
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. | ||
</p> | ||
|
||
<p> | ||
The subtleties and confusion around this question are captured by | ||
Wikipedia's | ||
<a href="https://en.wikipedia.org/wiki/Boy_or_girl_paradox" | ||
>Boy or Girl Paradox</a | ||
>. 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." | ||
</p> | ||
</div> | ||
</body> | ||
</html> |
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table { | ||
border-collapse: collapse; | ||
} | ||
|
||
th, td { | ||
border: 1px solid black; | ||
padding: 8px; | ||
text-align: center; | ||
} | ||
|
||
td.gg { | ||
background-color: lightpink; | ||
} | ||
|
||
td.bg { | ||
background-color: plum; | ||
} | ||
|
||
td.bb { | ||
background-color: lightblue; | ||
} | ||
|
||
.comment { | ||
margin: 10px; | ||
} | ||
|
||
figcaption { | ||
text-align: center; | ||
margin-top: 3px; | ||
color: slategray; | ||
} | ||
|
||
a { | ||
text-decoration: none; | ||
} |
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body { | ||
font-family: 'Roboto', sans-serif; | ||
background-color: #f5f5f5; | ||
margin-bottom: 100px; | ||
margin-left: 10px; | ||
} | ||
|
||
@media only screen and (max-width: 1000px) { | ||
body { | ||
margin-left: 10px; | ||
} | ||
} | ||
|
||
.content { | ||
max-width: 1000px; | ||
width: 100%; | ||
margin: 0 auto; | ||
} | ||
|
||
img { | ||
box-shadow: 0px 0px 3px 3px lightgray; | ||
} | ||
|
||
table { | ||
border-collapse: collapse; | ||
} | ||
|
||
th, td { | ||
border: 1px solid black; | ||
padding: 8px; | ||
text-align: center; | ||
} | ||
|
||
h1, h2 { | ||
text-align: center; | ||
} | ||
|
||
h2 { | ||
margin-top: 30px; | ||
} | ||
|
||
li { | ||
margin-bottom: 15px; | ||
} | ||
|
||
/* sort of a hack oh well */ | ||
li ul li { | ||
margin-top: 15px; | ||
} |