I was asked the question “a man states he has two children, and at least one of them are boys, what are the odds that the man has two boys?” I’ve been told the answer is 1/3, but I can’t wrap my head around it. Additionally, there is another version of the problem that states he has at least one boy born on a Tuesday. How does that change the odds? Why?
Edited to add (so people don’t have to sort through replies): the answer is 1/3 because “at least one boy” is accounting for B/G & G/B. The girl can be the first or second child. You can move the odds to 50/50 by rewording the question to “my first of two children is a boy, what are the odds the other child is a boy”
In: Mathematics
But…. there are two kids. One is a boy. So we can eliminate that from consideration. The question then becomes “there is one remaining child to consider. What are the odds that this child is a boy?” Therefore, one in two. Why isn’t this the answer?
Its a classic problem that I despise becuase it requires an additional bit of information that isn’t stated, and logically shouldn’t be assumed, but has to be assumed to get the answer to be 1/3.
That additional info is that despite being the father and despite having the knowledge that at least one child is a boy the father for some silly reason has no clue whatsoever which one of the children that is.
That is a necessary extra bit of info in order for it to not be 50%. The sensible interpretation of the meaning of the father saying “at least one is a boy” is that the father can point to one of the children and say “That one. That’s the one I know is a boy. I’m not sure about the other one.” That would mean “The gender of one child is locked down and thus no longer one of the variables. There is only one child with unknown gender left.”
The only way you get 1/3 is if the father meant “Oh, I know the gender of one of them but I still have no idea which of the two children is the one I know.” (Which is utterly bizarre and makes no sense at all as a way to picture the scenario.)
The “math class” answer is 1/3 for reasons that everyone else is explaining. In real life the answer is “it depends”. The difference is part of a subject of math called *Bayesian Inference*.
The *method* in which you find out information is just as important to the math problem as the actual information you learn.
The answer is 1/3 when those three combinations are equally likely. You might run into this situation during a meeting of the “has two kids” club and you ask a fellow parent “do you have at least one boy?” The question is usually constructed so that this is the most natural interpretation.
The answer is 1/2 in other situations. Say for example you are canvassing houses in the “has two kids” clubs, and one of the children answer, and it’s a boy. Assuming that boys and girls are equally likely to answer the door, the answer is 1/2 because BB is twice as likely to result in a boy answering a door than either of BG or GB. IMO this is generally a more common real-life situation.
The pieces of information that you learn are the same – but in probability it matters *how* the information is revealed to you.
This is also the key insight in the Monty Hall problem.
A man states he flipped a coin twice and at least one was heads. What are the chances he flipped two heads?
The Monty Hall problem doesn’t apply in this way. The order of birth has absolutely nothing to do with this situation. People are assigning values to this situation that don’t actually exist.
If we wanted to Monty Hall this situation, it would be that person has three children and we know one is a girl. We guess which one, in order, and then the parent tells us one of other two is a boy. It would benefit us to change our guess to the remaining child of those two, as there is a 2/3 chance that is a girl, as opposed to our initial 1/3 chance.
Alternatively, we could Bertrand’s box it. In that case there are three parents, each with two children. One has two boys, one two girls, and one a boy and a girl. If one of those parents revealed the sex of one child at random, telling us that the child is a girl, there is a 2/3 chance the other is also a girl.
The question is notoriously ambiguous.
If i have a son, and i say to you “hey this is my son timmy. I also have another child. Do you think it’s a boy or a girl?” The probability you guess right is 1/2. You don’t know which brother is timmy, but *i do*. It is my family, and i gave you a precise information, you just don’t know it. If i know timmy is the oldest i’m actually asking the probability my family is boy-girl or boy-boy. I’m not throwing girl-boy into the mix because i know timmy is my older son, despite i haven’t told you. Conversely, if i know timmy is the youngest i’m just asking girl-boy vs boy-boy. You don’t know which question i’m asking you, but it doesn’t matter because the answer is 1/2 to both. I’m basically just asking you to guess the gender of a person you don’t know, in a fancy way. The answer is in fact 1/2 but this is NOT what your question is asking.
On the other hand your question is more like this: consider all the families in your hometown with exactly two children. I choose one at random and go to their house. I only see one boy – don’t know if it’s the older, younger, and haven’t seen the other child. Based on that, what’s the probability i chose a family with two boys? Well, there are 4 ways a 2-children family can be: GG, BB, BG or GB. Suppose they are all equally likely. Can’t be GG because i saw the boy, but i don’t know the family, so it’s equally likely i saw the older sibling of a BG family, the younger of a GB family, or either one of a BB family. Putting the probability of BB at 1/3.
It is a bit more complex than your edit because the question is a bit deceptive. Most means of knowing that one of a man’s two children is a boy would make it 50/50 on the other being a boy. For example, meeting one of the boys, the man mentioning his son did something, etc. One of the few ways to get this scenario is to ask the man with two children if he has any sons, and he says yes. What are the odds he has no daughters? 1/3. Another way is to know a king has two children, and he has an heir, and heirs can only be male. The odds he has two sons is 1/3. Or, you meet the heir to a king, and he has 1 sibling. The odds are 1/3 that sibling is a boy.
Most natural ways of finding out someone has a son makes it such that there is a 50/50 chance of them having two sons if they have two children because them mentioning a son sort of switched the perspective to being you know a boy has a sibling. What are the odds that sibling is a girl? Well, 50% of boys have a sister and 50% have a brother even though only 2/3 of families with a boy have a boy and a girl while only 1/3 are boy boy.
It’s because the two children are considered “distinguishable.” So there is a difference between BG and GB. If the two were indistinguishable, then the odds would be 50:50 that it’s either two boys or a boy and a girl.
This reminds me of a talk on some podcast about how lotto companies give money to schools so that probability isn’t taught at the appropriate levels
From the original story…
Fact: two children
Fact: one child is a boy
Proposition: what is other child?
It’s 1/2 as there is only one child under question.
Introducing an order is introducing an extra complication which, in this case, is not stated in the original statement.
Latest Answers