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Q&A

How would you intuit, and soothsay to rewrite, "both girls, ≥ 1 winter girl" as "both girls, ≥ 1 winter child"?

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  1. Please see the red underline. How can you intuit that "both girls, at least one winter girl" as "both girls, at least one winter child"? I ask this for my 15 y.o.

  2. This step feels fey, sibylline! How would you prognosticate to use this fact? We would've never augured to construe "both girls, at least one winter girl" as "both girls, at least one winter child"!

Adam Bailey's answer details the calculations.

Example 2.2.7 (A girl born in winter).

A family has two children. Find the probability that both children are girls, given that at least one of the two is a girl who was born in winter. In addition to the assumptions from Example 2.2.5, assume that the four seasons are equally likely and that gender is independent of season. (This means that knowing the gender gives no information about the probabilities of the seasons, and vice versa; see Section 2.5 for much more about independence.)

Image alt text

At first this result seems absurd! In Example 2.2.5, the result was that the conditional probability of both children being girls, given that at least one is a girl, is 1/3; why should it be any different when we learn that at least one is a winter-born girl? The point is that information about the birth season brings at least one is a girl" closer to a specific one is a girl". Conditioning on more and more specific information brings the probability closer and closer to 1/2.

For example, conditioning on at least one is a girl who was born on a March 31 at 8:20 pm" comes very close to specifying a child, and learning information about a specific child does not give us information about the other child. The seemingly irrelevant information such as season of birth interpolates between the two parts of Example 2.2.5. Exercise 29 generalizes this example to an arbitrary characteristic that is independent of gender. □

Blitzstein. Introduction to Probability (2019 2 ed). pp 51-2.

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