De-Paradoxing The Boy/Girl Paradox

February 12, 2023234 Comments

The boy/girl paradox, also known as the two child problem, is a problem/puzzle in probability theory that goes something like this:

  • Mr. Smith has two children. The older child is a boy. What is the probability that both children are boys?
  • Mr. Jones has two children. At least one of them is a boy. What is the probability that both children are boys?

The answer to the first question is 1 in 2. The answer to the second question is often said to be 1 in 3 (although it could be 1 in 2 as we will see later). But how can knowing “at least one of them is a boy” as opposed to “the older child is a boy” affect the probability? There is an apparent paradox here.

But, before we can make sense of this paradox, we need to be clear about what exactly we mean when we talk about “probability”.

What does “probability” mean?

Suppose you and I meet Mr. Smith and learn the his older child is a boy. You and I can agree that the probability of his other child being a boy is 1 in 2. But Mr. Smith certainly would not agree – he knows the sex of his other child so he would either say the probability is zero or it is 100 percent. And he would be correct. Does that mean we were incorrect in our conclusion that the probability is 1 in 2? Let’s see….

Let’s consider a coin toss. We can agree before the toss, or even while the coin is in the air, that the probability of it landing heads-up is 1 in 2.

But what if I let the coin land and cover it with my hand before either of us has seen it. Is the probability still 1 in 2? The coin has already landed one way or the other, so what do we mean when we say there is a 1 in 2 chance that it is heads? And what if we reveal the coin a few seconds later and find out that it is tails? Does that mean that we were wrong a few seconds earlier in saying that there was a 1 in 2 chance that it was heads, when in fact it was tails all along?

Or what if we toss and cover 2 coins? We would say that the probability of both being heads is 25% (1 in 4). But what if we let a some friends see one of the coins (but not reveal to us what they see) – they would agree among themselves (without telling us) that the probability of both being heads is either zero or it is 50% (1 in 2). Meanwhile, you and I would still be saying the probability is 25%. We can’t all be correct, can we?

But, all paradoxes disappear if we think of the probability of a statement being true as meaning “the proportion of times we are in this situation that it will turn out that the statement is true“:

    > Coin already landed: it might in fact already be tails, but we haven’t seen it yet: 50% of the time we are in this situation it will turn out to be heads. No paradox.

    > Two coins covered: We are in the situation where we do not know how either of the coins have landed. Our friends have seen one but have not given us any indication of what they have seen. 25% of the time we are in this situation it will turn out that both are heads. Our friends, on the other hand, have seen one of the coins. In their situation, it will turn out that both are heads either 50% or 0% of the time (depending on whether the coin they saw was heads or tails). No paradox.

    > And going back to Mr. Smith: the situation we are in is we’ve met a man who has two children, and we know the older one is a boy. We haven’t asked him yet about the other child and he hasn’t told us. 50% of the time we are in that situation it will turn out that both are boys. Mr. Smith, on the other hand, is in the situation where he knows for certain whether the other child is a boy or a girl. The presence of two maths nerds discussing probabilities doesn’t change that. No paradox.

    But, what does “in this situation” mean?

    So all we need to do is work out what proportion of times we are “in this situation” a particular statement will turn out to be true. But we’re not out of the woods yet! We need to carefully consider what exactly we mean by “in this situation”.

    Imagine a town with 40 opposite-sex couples, with a strict 2-child policy. If each birth is equally likely to be a boy or a girl, then we would expect something close to the following:

    • 20 couples’ first child will be boy and 20 couples’ first child will be a girl.
    • Of the 20 who have a boy first, 10 will go on to have a another boy, while 10 will go on to have a girl.
    • Of the 20 who have a girl first, 10 will go on to have a boy, while 10 will go on to have a another girl.

    The result will then be:

    • 10 couples with 2 boys
    • 10 couples with 2 girls
    • 20 couples with a boy and a girl (10 who had the boy first and 10 who had the girl first)

    Now, imagine the town organises an event for all the boys and they are accompanied by their fathers. At this event there will be 30 men:

    • 10 men who have 2 sons
    • 20 men who have a son and a daughter.

    So if you visit this event and randomly meet a man (who you know has at least one son, as he is at the event), there is a 1 in 3 (10 in 30) chance that he has 2 sons.

    So it looks like we have the answer to the Mr Jones question posed at the beginning. Not quite – it depends on how you meet Mr Jones.

    Remember that also at this event are the sons of the 30 men:

    • 20 boys who are the sons of the 10 men who have 2 sons
    • 20 boys who are the sons of the 20 men who have a son and a daughter.

    So, if you randomly meet a boy at this event, there is a 1 in 2 chance that he comes from a 2-son family. If he then introduces you to his father, then the probability that the man has 2 sons is now 1 in 2, not 1 in 3.

    So: if you meet a man a this event, the the probability that he has two sons depends on the situation (that is, it depends on how you meet him):

    • If you randomly walk up to a man in the crowd, then there is a 1 in 3 chance that you will pick a man with 2 sons.
    • If you randomly pick a boy and ask to be introduced to his father, then there is a 1 in 2 chance that you will end up meeting a man with 2 sons (the 10 men with 2 sons have “two tickets” in the draw).

    In the above situations where you deliberately select a man or a boy, it is clear which situation you are in. But in other cases, you might have to put a bit of thought into it.

    • If there is an sporting activity that all the boys are taking part in, and all the men are watching, and you happen to get talking to one of the men in the crowd, then is is clear that all the men had an equal change of being the one you got talking to, so in that situation there was a 1 in 3 chance that you would end up talking to one of the men who has 2 sons.
    • But if, say, you happen to be outside the medical centre at this event and you meet a man waiting for his son to come out after receiving treatment for an injury. In this case, you didn’t really select a man at random. Instead, the sporting activity randomly selected a boy to be injured and this caused you to meet his father, so this situation is similar to the one above where you randomly selected a boy. Another way to look at it is that each man with 2 sons was twice as likely to end up bringing his son to the medical centre (“two tickets in the draw”). In this situation there was a 1 in 2 chance that would you end up talking to one of the men who has 2 sons.

    Earlier we said that the probability of a statement being true is “the proportion of times we are in this situation that it will turn out that the statement is true”. Now we see that you have to be very careful to make sure you correctly define “in this situation“.

    What was the question, again?

    The original questions were:

    • Mr. Smith has two children. The older child is a boy. What is the probability that both children are boys?
    • Mr. Jones has two children. At least one of them is a boy. What is the probability that both children are boys?

    We can safely say that in any normal situation we can think of regarding the meeting of Mr Smith, the chance of him having 2 sons is 50%.

    As for the meeting of Mr. Jones: well as we saw above, the answer could be 1 in 2 or 1 in 3, depending on the situation. The question as stated above doesn’t describe any situation at all, so we can’t say which one it is. We can only assume that the question setter meant “if you randomly select a man from all the men with 2 children, at least one of whom is a boy, what it the probability that both the children are boys?” In this case the answer is 1 in 3.

    But let’s see if we can pose the question with a clearly defined situation. How about this:

    • You are walking down the street and you happen to meet a man walking with his son. You get talking to him and he tells you that he has two children. What is the probability that both children are boys?

    To answer this, we need to determine what proportion of times we are in this situation that it will turn out that both children are boys.

    The crux of the question is what do we mean by “in this situation“? Or “how did we end up talking to this particular man”?

    • Did we randomly select the man?
    • Or did we randomly select the boy? In other words, if the man had 2 sons, were we twice as likely to bump into him?

    To clearly understand what we mean by “in this situation”, imagine we wanted to measure the probability experimentally and decided to deliberately get into this situation. How would we go about it?

    • We could walk down the street until we meet a man walking with his son. If we see a man walking with a daughter, or on his own, or with more than one child, or if we see a woman, then we walk on. When we meet a man with a son, we stop and ask how many children he has. If the answer is 2 then he is included in our experiment. In this situation, the probability that the other child is also a boy is 1 in 2. This is because, although a man with 2 sons isn’t twice as likely to be out walking, he is twice as likely to be included in our experiment.
    • We could say that the question doesn’t actually have to be about 2 sons. We would say we stop and talk to the first parent we see walking with one child, ask how many children he or she has and if the answer is 2, we try to determine the probability that the other child is the same sex. In this case we can see that there is no bias towards boys or girls, so the answer must also be 1 in 2.

    In both cases the answer is 1 in 2, not 1 in 3. And in a lot of normal situations, it will 1 in 2, not 1 in 3. Let’s look at why this is so:

    As we saw earlier, if we assume that each birth is equally likely to be a boy or girl, we can expect the following 4 groups of two-child couples to be roughly of equal size.

    • couples with 2 girls
    • couples with 2 boys
    • couples who had a boy first and then a girl
    • couples who had a girl first and then a boy

    If we could simply eliminate the first group and then randomly select a parent from the remaining couples, we would have a 1 in 3 chance of selecting a parent with 2 boys. But how do we eliminate the first group? Well, a lot of times this happens when we find out about one of the boys – we meet or see the boy. But, in those situations, we haven’t really randomly selected a parent. Instead we have randomly selected a boy. Or rather, circumstances have randomly selected a boy for us to meet or see or become aware of in some way. To put it another way, each of the couples with 2 boys had “two tickets in the draw”, thus increasing their chances of being selected. In these situations, the chance of them having 2 boys is 1 in 2.

    The probability is 1 in 3 only in situations where the parent you are talking to can be eliminated from the first group without you becoming aware of either of their individual children. An example might be if you meet a parent at an information night at an all-boys school. Or you meet a parent at that previously mentioned event attended by all the boys in the fictional town (in real life, however, if there is a sporting event for boys, a parent with 2 boys is twice as likely to be there, so the chances would actually be 1 in 2).

    Conclusion

    In most situations where you become aware that a parent of two children has at least one son, it is because you have become aware of one individual who is a son. It doesn’t matter whether you know that he is the oldest – the pertinent fact is you have become aware of one individual son. Knowing “the one who was out walking today is a boy” is similar to knowing “the older one is a boy“. In both situations, the probability of the the other child being a boy is 1 in 2.

    Main Points

    > Problems in probability are easier to understand and paradoxes tend to disappear if we think of the probability of a statement being true as meaning “the proportion of times we are in this situation that it will turn out that the statement is true“.

    > The crux of the question is what do we mean by “in this situation“?

    > To determine what is meant by “in this situation”, think about how you would design an experiment to get into the situation – what events are necessary to get into the situation and what influences the chances of those event occurring?

    Links

    Here’s a few links to other probability problems:

    Frog riddle (Youtube):
    https://www.youtube.com/watch?v=cpwSGsb-rTs
    (Note that using the reasoning in this post will result in an answer that contradicts the answer presented in the video. See which answer you think is correct).

    Sleeping Beauty Problem (Wikipedia):
    https://en.wikipedia.org/wiki/Sleeping_Beauty_problem

    And here’s a couple of links to the Boy/Girl paradox discussed in this post:

    Wikipedia: https://en.wikipedia.org/wiki/Boy_or_Girl_paradox

    Guardian: https://www.theguardian.com/science/2019/nov/18/did-you-solve-it-the-two-child-problem

    Tags:
    Last updated on February 14, 2023

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      – Заметное улучшение текстуры кожи
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