Birth order isn’t important by itself. What is needed is a way of differentiating between two children that doesn’t depend on gender. You can choose any binary characteristic as long as it is distinct and unambiguous. Birth order is the easiest, but others will do and provide the exact same results. And when you introduce a non-binary characteristic (weekday of birth, month of birth) you get different results.
Again, it’s possible to create a word problem where these things don’t factor in, but for the word problems presented here they are important and make the results non-intuitive.
Yeah. I think there is an element of following a pattern with some amount of blindness to the underlying need.
Everyone knows the mantra, if we start with a pair of dice, people know the drill to work out the set of possible throws. and then the probabilities of the various totals. One way of getting doubles, and two ways of getting each of the other combinations.
So the same approach is done on the question of how to set up questions about two children. One way of getting kids of the same gender, two ways of having one of each. Somehow, this triggers an innate idea that the “two ways” must clearly be due to birth order, and only due to birth order. Which takes on a defining characteristic that clouds reasoning about the probabilities.
Maybe this comes from parents categorising families this way. No idea how they manage three child families.
Exactly. This is the equivalent question variations:
I flip 2 coins - what are the odds at least one is heads?
(What are the odds exactly one is heads, the other is tails?)
I flip 2 coins. I pick a random one of the two; it is heads. What are the odds the other is also heads?
I flip a coin. It’s heads. What are the odds when I flip another coin, it too will be heads?
The answer depends on the information.
The children could be twins or separate gestations. (and we simlify the question by assuming 50-50 chance of boy or girl, and ignoring the skewed odds for genders and same genders of twins, etc. etc.)
So age or birth order is totally irrelevant. The key point is that there are two independent events, and when you are given a piece of information, it could be for one event or the other event - you don’t know which…
Ignoring the odds of fraternal vs. identical twins, assuming 50-50 odds for each child, consider:
If Sally has twins, and you knock on the door and a girl answers, what are the odds the other child is a girl?
The puzzle boils down to - only information you have is:
There are two children.
One is a girl.
Odds of boy vs. girl for each child when born is 50-50.
This information only eliminates the BB case.
Where birth order might be relevant is knowing that Sally had a girl, then had another child. Or that Sally had her first child, then had a girl. What are the odds for the other child in either case?
This information in each case tells you also which event - eliminates Bx in the first case, and xB in the second.
Yeah, no, I’m sorry. I still don’t get this (and possibly also several other variations):
2. Mrs Jones has two children. At least one is a boy. What’s the chance that both are boys?
If at least one is a boy, there are three possible equally likely gender-assignations of two siblings. boy-boy, boy-girl, or girl-boy. Only 1 in 3 cases, or 33 per cent are both boys. The lesson here is that when considering equally likely scenarios we must consider birth order. If the birth order of the boy is not specified – i.e. if we don’t know if he is the eldest or the youngest – the probability of two boys drops to 1 in 3.
I do not see how that question calls for order of birth to be relevant in the answer and why the answer should be anything but 1/2.
The order of birth is only relevant in distinguishing the boys, but is not necessary.
In this case, you first pick a pair: the children of Mrs Jones. At that point, you know there are two children, and that there are three possibilities: two boys (25% chance), two girls (25% chance), or one of each (50% chance).
Next, we learn at least one is a boy. That eliminates the two-girl possibility. The remaining possibilities add up to 75%, so we need to renormalize based on that. Which gives two boys is 33% (= 25% / 75%), and one of each 67% (= 50% / 75%).
Birth order is how you distinguish between these two cases. Remember, the question is about the SET of two children, not about the remaining child since there’s no way to know which is the remaining child from the set.
Another way of phrasing this is imagine two coins, a nickel and a quarter. Before we start I tell you that I’m going to flip both, look at them, and if at least one is tails, I will tell you that at least one is tails. You’ll then tell me the likelihood that the other coin is also tails.
Rather than birth order, we’re using different coins to distinguish the two items apart. There are four equally likely possibilities:
Nickel (heads) Quarter (heads)
Nickel (tails) Quarter (tails)
Nickel (heads) Quarter (tails)
Nickel (tails) Quarter (heads)
If I tell you that one coin came up tails (eliminating choice #1) then there’s only a 1/3 chance that the other coin is tails, right? Do you understand and agree with this analysis?
If so, you just need to work on understanding how this is equivalent to the two kids problem. Again, all that matters is that we have a way to distinguish between the two items. Types of coins, birth order, it doesn’t really matter what as long as it’s binary and conclusive.
Birth order is just a convenient way to completely distinguish between “child A” and “child B” that is agnostic to gender, in a way that is universal to all parents with 2 children (so you could, for example, ask a stadium full of 1000 parents with 2 children the same question and you’d expect to get an answer)
It doesn’t have to be birth order, it could be “the one that answered the front door when I knocked on it/ the one that didn’t*” , or “the one that was picked on a random coin flip / the one that wasn’t”
starting out with “at least one of the two children is a boy”, you only have the three valid scenarios with 1/3 chance each (since the girl-girl scenario is eliminated), two of which contain a girl:
A ____B
boy -boy
girl - boy
boy - girl
If you were to ask 1000 parents of two-children “how many of you have at least one boy?”, you’d expect ~750 parents to raise their hands, only 250 of which (1/3 of 750) are parents of two boys.
If you asked 1000 parents of two-children “how many of you have a first-born boy?”, you’d expect ~500 parents to raise their hands, 250 of which (1/2 of 500) are parents of two boys.
If you asked 1000 parents of two children to assign heads & tails to each child and then flip a coin, and then ask “how many of you have a heads boy?”, you’d also expect ~500 parents to raise their hands, 250 of which (1/2 of 500) are parents of two boys.
*assuming no gender bias in who opens the door first.
Incorrect. You eliminate the case where the one who doesn’t answer the door is B and the one who does answer the door is B, and you also eliminate the case where the one who doesn’t answer the door is G and the one who does answer the door is B.
What it eliminates depends on how we learn that at least one is a boy.
But the only way that learning one is a boy eliminates only the two-girl possibility is with an unclear lawyerly interpretation. In the vast majority of ordinary situations, learning that at least one is a boy will happen in a way that eliminates two possibilities, and BG and BB are equally likely.
Given that substantially all these things are contrived gotcha situations, IMO lawyering the wording is EXACTLY what the problem is really all about. The math is trivial once you correctly determine what the question is. But the question can only be determined correctly by very careful analysis of the wording.
Further quite often these things get passed around and subtle changes slip into the wording that the various cut-n-paste “authors” don’t bother to recognize, much less analyze or better yet fix. Leading to wrong answers being labeled right which further confuses the folks trying to learn something at home.
I’d expect a question in a math textbook to be pretty lawyering-free. And to have no C&P mistakes. Something making the rounds on TwitFace or some puzzle website? You’re going to need Perry Mason and Clarence Darrow to have a good chance of getting it right. And only then if the “author” didn’t screw it up.
But we’re not in an ordinary situation, we’re in the context of my post and the post I was quoting. (I know you wouldn’t quote me out of context.)
“Mrs Jones has two children”–that seems like a straight-forward fact.
“At least one is a boy”–that also seems like a straight-forward fact.
In the usual math-problem context, I interpret these statements with the minimum assumptions needed to make them true. Eliminating a “boy-girl” possibility but not a “girl-boy” one (or vice versa) is not a good interpretation in this context.
Maybe you read math problems differently than I do, and I’m curious to understand how.
Like others have said, those ordinary situations are not interesting wrt this probability problem. If they wanted to ask you what the likely outcome of a single coin flip was, they would ask that question. These word problems are designed (and if implemented properly) to ask more interesting questions.
But it’s not, because how we learned that fact matters.
And the minimum assumption is that we learned that “at least one is a boy” by gaining knowledge about one specific child (by meeting him, for instance), because that’s by far the most common way of gaining that information.
Hmmm, yes, I think I see it now. “In the wild,” so to speak, half of all two(-and only two)-sibling families (excluding those with non-binary siblings) will be a boy and a girl. So if, in the wild, one spots a sibling from such a family, odds are already stacked in the favor of their other sibling being opposite-gendered. Because we’re twice as likely to spot members of a mixed-gender pair in the wild as we are to spot members of a boy-only pair, and we’re also twice as likely to spot members of a mixed-gender pair as a girl-only pair. Spotting a boy eliminates the girl-only pair from consideration, but doesn’t change the fact that there are twice as many mixed-gender pairs as there are boy-only pairs.
Children: like wild ducks, but you can’t shoot 'em, am I right?
Actually, I should note, I think it only holds true if only one member of a sibling pair is permitted to run wild at a time. Which is, I suppose, why the “answering the door” setup works, as one might tend to assume that only one child would answer the door at a time. Because of course otherwise, there would be two boys for every boy-boy pair running around in the wild (and so while there might be only half as many boy-boy pairs as mixed-pairs, there would be an equal number of boys to spot from either boy-boy or the doubled number of mix-gendered pairs).