Moms’ Talk

Two moms, Sarah and Courtney, are talking to each other.

Sarah: I have two children.
What is the probability that both of Sarah’s children are boys?

Courtney: Me too! Do you have any boys?
What is the probability that both of Courtney’s children are boys?

Sarah: Yes, I do! What is your younger child?
What is the probability that both of Sarah’s children are boys?

Courtney: It is a boy. He is so mischievous!
What is the probability that both of Courtney’s children are boys?

Sarah: Is he Sagittarius? Sagittarius boys are known to drive their mothers crazy. I can testify from personal experience.
What is the probability that both of Sarah’s children are boys?

Courtney: No, but actually I have the opposite personal experience to yours.
What is the probability that both of Courtney’s children are boys?

Sarah: Well, I guess astrology does not always get it right.

Courtney: I assume it does about half of the time.

The answers are: ~1/4, ~1/4, ~1/3, ~1/2, ~23/47, 1.

Explanation:

Initially, we do not have any information about the children and therefore the chance that both of them boys is 1/2 × 1/2. This applies to the first and the second question.

After Sarah says that she has at least one boy, there are equal possibilities that she has Boy + Boy, Boy + Girl, or Girl + Boy. Therefore, the chance that both children are boys is 1/3.

After Courtney says that her younger child is a boy, the only remaining question is what is the gender of her older child, and therefore the chance is 1/2.

The fifth exchange implies that Sarah has a Sagittarius boy. There are 23 combinations such that both children are boys and at least one of them is Sagittarius. There are 47 combinations such that at least one of the children is a Sagittarius boy. Therefore, the chance that both children are boys is 23/47.

Finally, Courtney says that her younger child, which is a boy, is not Sagittarius, but her personal experience with Sagittarius boys is positive. Therefore, her older child is a Sagittarius boy and the chance is 1.

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  1. This is wrong. Age is irrelevant to if someone is a boy or a girl.

    If you know someone has two kids, and at least one is a boy, then you know there is a 50/50 chance that the other unnamed child is a boy or girl. It doesn’t matter if it’s a boy, girl combination or a girl then a boy. both are just girl, boy vs boy boy.

    By that logic if someone was to flip a coin twice, and you knew one was a heads, you could ‘logically’ deduce a 33% chance both were heads because you could have tails heads, heads tails or heads heads.

    Order doesn’t come into this.

    1. Hello Dale, thank you for your response.
      The formal way for deriving these probabilities is by using the formula for conditional probability.

      P(A given B) = P(A and B)/P(B)

      In this case, A is the event that both children are boys. B is the event that the younger child is a boy or that there is at least one boy, respectively. Since having two boys implies that the younger child is a boy, which in turn implies that there is at least one boy, in this problem we get:

      P(A given B) = P(A and B)/P(B) = P(A)/P(B)

      Now, it is pretty clear that the probability that the younger child is a boy is different (smaller), from the probability that there is at least one boy. The former is 1/2, whereas the latter is 3/4. Since P(A) = 1/4, the two answers should be 0.25/0.5 = 1/2 and 0.25/0.75 = 1/3 respectively.

      This is the so called “Boy or Girl Paradox”. To read more about it, check the Wikipedia page.
      https://en.wikipedia.org/wiki/Boy_or_Girl_paradox

      1. I read through the wiki on this subject and at best it is a trick question that isn’t actually about math but about poor question wording (something the article discusses) but at worst is completely wrong. Further down it explicitly states that if you are given the line of “at least one boy” then the probability is .5.

        1. Hello Devon. At its essence, this puzzle is a combination of several strictly math (probability) questions:

          1) If you have 2 children and the older one is a boy, what is the chance that the younger one is a boy? Answer: 50%

          2) If you have 2 children and at least one is a boy, what is the chance that both are boys? Answer: 33%

          3) If you have 2 children and at least one is a boy born in January, what is the chance that both are boys? Answer: 49%

          Of course, the wording of the puzzle is more intricate, so that it fits within a real-life conversation, but the problems listed above are well defined and have precise answers. I created a topic about this puzzle in our forum, so if you like, we can discuss it further there:)

          https://www.puzzleprime.com/forums/topic/moms-talk/