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

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2024-11-16 21:30:03

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.

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.

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

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.

You know as soon as the boy answers that you're twice as likely to be talking to row 4 column 1 as row 2 column 1. But there are twice as many BG families as 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.

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