The family with two children paradox is one of most confusing puzzles to be asked during the interviews. Here we listed a collection of the common puzzles.
1. Consider a family that has two children. We are interested in the children's genders. Our sample space is S={(G,G),(G,B),(B,G),(B,B)}. Also assume that all four possible outcomes are equally likely.
What is the probability that both children are girls given that the first child is a girl?
We ask the father: "Do you have at least one daughter?" He responds "Yes!" Given this extra information, what is the probability that both children are girls? In other words, what is the probability that both children are girls given that we know at least one of them is a girl?
2. If you are not yet confused, let's look at another family-with-two-children problem! I know that a family has two children. I see one of the children in the mall and notice that she is a girl. What is the probability that both children are girls? Again compare your result with the second part of Example 1.18. Note: Let's agree on what precisely the problem statement means. Here is a more precise statement of the problem: "A family has two children. We choose one of them at random and find out that she is a girl. What is the probability that both children are girls?"
3. A family has n children (n > 2). We ask the father: "Do you have at least one daughter?" He responds "Yes!" Given this extra information, what is the probability that all n children are girls? In other words, what is the probability that all of their children are girls, given that at least one of them is a girl?
4. A family has n children (n > 2). We ask from the father, "Do you have at least one daughter named Lilia?" He replies, "Yes!" What is the probability that all of their children are girls? In other words, we want to find the probability that all n children are girls, given that the family has at least one daughter named Lilia. Here you can assume that if a child is a girl, her name will be Lilia with probability α≪1 independently from other children's names. If the child is a boy, his name will not be Lilia.
5. A family has n children. We pick one of them at random and find out that she is a girl. What is the probability that all their children are girls?
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