Greetings to all.
I started an MSc that includes a course in statistics. Full disclosure: my bachelor’s had no courses of statics and it is in biology.
So, the professor was trying to explain the Bayes theorem and conditional probability through the following example.
“A friend of yours invites you over. He says he has 2 children. When you go over, a child opens the door for you and it is a boy. What is the probability that the other child is a boy as well.”
The math say the probability the other child is a boy is increased the moment we learn that one of the kids is a boy. Which i cannot wrap my head around, assuming that each birth is a separate event (the fact that a boy was born does not affect the result of the other birth), and the result of each birth can be a boy or a girl with 50/50 chance.
I get that “math says so” but… Could someone please explain? thank you
In: 77
In the ‘boy or girl’ problem (where you imagine 2 child familes with either daughters or sons), you get some information about one child, and then try to work out the probability of the other child’s gender.
An important thing here is that it is very very important *how* you come about the information.
For instance, if you poll 2-child families and ask “Do you have a son named Bob?”, and get a ‘yes’, you might calculate a different probability than if you get into a conversation with them and they happen to mention their son Bob. (I forget if that precise difference in circumstance changes our probability calculation, but that *sort* of nuanced and subtle change can matter.)
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Thinking about your example, the exact way in which you learn they have at least 1 boy matters.
Maybe you show up to their house and see that a son opens the door. (Which is the scenario your professor said.)
But note that if they fill out a survey that asks “Do you have 1 or more sons? Y / N” and answer Y; or if they mentioned before you visit “My son Bob loves football and was born on a Sunday.” then that might change the probability you should guess for their other child.
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I think in the wording you phrased, assuming that either child could have opened the door with equal probability, the professor made a mistake. The other kid has a 50-50 chance of people a boy or girl.
The answer might not be the same if the way you gain that information is slightly different.
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Here are two videos on the topic:
[https://www.youtube.com/watch?v=bDZieLmya_I&t=0s&ab_channel=ZachStar](https://www.youtube.com/watch?v=bDZieLmya_I&t=0s&ab_channel=ZachStar)
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