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# Two boys

## Question

I ask people at random if they have two children and also if one is a boy born on a tuesday. After a long search I finally find someone who answers yes. What is the probability that this person has two boys? Assume an equal chance of giving birth to either sex and an equal chance to giving birth on any day.

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Spoiler

If we map out all the possible 'day of the week' combinations on 4 separate grids of days, we get the following totals where at least one boy is born on a Tuesday:

B/G = 7

G/B = 7

B/B = 13

G/G = 0

Overall, there are a grand total of 27 possible outcomes.  The probability of both of the children being boys is then simply 13/27.

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Alas, English.... But I am confused. From your interviewing you know for a fact that this family has A boy. So regardless of their birth order all we don't know is the probability that the other child is a boy. And the other child could be a boy or girl, so I still believe it is 1/2.

The subtlety is…

Spoiler

…that you can only say that "the other child" has a ½ chance of being a boy when you know that one child specifically is a boy. For example, if I tell you that I have two children and my eldest child is a boy, then you can say that the probability that both of my children are boys is ½. However, if I merely tell you that I have two children and at least one is a boy, but do not in any way distinguish which child is definitely a boy, then all I have told you is that my children are not both girls. Three equally likely possibilities remain: the eldest might be the only boy, the youngest might be the only boy, or both might be boys.

In @bonanova's puzzle, the children are distinguished from one another if only the boy was born on a Tuesday (I do not know if that child is the eldest or not, but I can speak of "the child born on Tuesday" separately from "the other child"). If both children were born on Tuesday, then I cannot say "this one is a boy and the other one could be a boy or girl," because I have no means of distinguishing one child from the other.

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Spoiler

Having two children, there are four equally likely combinations of sexes.

If both children were born on Tuesday, then all we know for certain is that they are not both girls, which eliminates just one of the four cases. So then the probability that both are boys is ⅓.

If not, then we know the child who was born on Tuesday is a boy, which eliminates two of the four cases, and so the probability that the other child is also a boy is ½.

Since the probability of both children being born on Tuesday (given that at least one of them is) is 1/7, the overall probability that both children are boys is

(1/7)*(1/3) + (6/7)*(1/2) = 10/21.

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@ThunderCloudThat's close, but a bit low.

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5 hours ago, bonanova said:

@ThunderCloudThat's close, but a bit low.

Hmmm… aha!

Spoiler

Let's temporarily label the children first-born and second-born. For any parent with two children, there are 49 possibilities for what days of the week their first-born and second-born child had a birthday. In 13 of them, at least one child is born on Tuesday; both children are born on Tuesday in only 1 case.

So! Given that at least one child was born on Tuesday, the probability that both are born on Tuesday is then 1/13. If both are born on Tuesday, then we know they aren't both girls, so the probability that both are boys is ⅓. If one of them wasn't born on Tuesday, then that child has a ½ probability of being male.

Overall, then, the probability is (1/13)*(1/3) + (12/13)*(1/2) = 19/39 that both children are boys.

That should do it. ^^

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On 4/10/2018 at 3:38 AM, bonanova said:

I ask people at random if they have two children and also if one is a boy born on a tuesday. After a long search I finally find someone who answers yes. What is the probability that this person has two boys? Assume an equal chance of giving birth to either sex and an equal chance to giving birth on any day.

If it is given that they already have one boy then...

I get 1/2. Since the second child could either be a boy or a girl.

Note: The spoiler buttons seems to be missing on mobile again.

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@ThunderCloud you're homing in on it, but now you're a little high.

@BMAD It's certainly true that if the FIRST child was a boy born on a tuesday, then it's just the prob that the second child is a boy. But ... the OP does not tell you that. That is, "one is a boy" does not imply "my oldest child is a boy." So your "second" child simply means the "other" child.

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4 hours ago, bonanova said:

@ThunderCloud you're homing in on it, but now you're a little high.

@BMAD It's certainly true that if the FIRST child was a boy born on a tuesday, then it's just the prob that the second child is a boy. But ... the OP does not tell you that. That is, "one is a boy" does not imply "my oldest child is a boy." So your "second" child simply means the "other" child.

Alas, English.... But I am confused. From your interviewing you know for a fact that this family has A boy. So regardless of their birth order all we don't know is the probability that the other child is a boy. And the other child could be a boy or girl, so I still believe it is 1/2.

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ah, thank you.

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