Yes, I found this one easy. Was surprised my data management intuition came back after all these years since school. There’s really only three options:
- boy - boy
- boy - girl
- girl - girl
So it must be 1/3 chance. If you’re looking at permutations in order, that’s a different question.
This intuition is wrong even if turned out to get the right answer. The three unordered options do not have equal probabilities, boy+girl is twice as likely to occur as boy+boy and girl+girl.
To get the right answer you must be careful about conditional probabilities (or draw out the sample space explicitly). The crux of the issue is that you are told extra information, which changes your estimate of the probability.
(This question as written is very easy to misinterpret. The Monty Hall problem, which illustrates the same thing, is better since the sample selection is much more carefully explained.)
Oddly, this is a part I'm sticking with on this problem.
Specifically, if you know that one is a girl, then the unordered options seem like they are back on equal footing? That is, it isn't twice as likely if you know that one ordering can't happen? (Or, stated differently, you don't know which version of two girls you are looking at.)
So, for this one, you know that either the youngest is a girl (so, girl-boy is not possible) or that the oldest is a girl (so boy-girl is out). That puts you back to the rest of the possibilities. Boy-boy is out, sure, as you have a girl. But every other path remains? So, you have one of (boy-girl(known), girl-girl(known), girl(known)-boy, girl(known)-girl). Which drops you back to 50/50?
Like others in the thread have said, the question could have been phrased more precisely. Technically you are misreading it but in an annoying and trivial way.
What the problem is really saying is this:
1) You have a large collection of families with two kids of varying genders.
2) You draw one of them at random. At this point, your only estimate of P(2 girls) is 0.25.
3) Someone tells you that the family you drew has at least one girl.
4) This extra information changes your probability estimate because the possibility of two boys has been ruled out; the naive 1/4 estimate is refined to 1/3.
The way you are interpreting it is this:
1) You have a large collection of families with two kids, at least one of whom is a girl.
2) Then the probability that the other child is a girl is clearly 50%.
As a reminder this is how the original post phrased the question:
Here's the problem: a family has two children. You're told that at least one of them is a girl. What's the probability both are girls?
This is just too vague and admits both interpretations, they needed to be more specific about where the family "came from." That's why Monty Hall is a better illustration: it starts with you explicitly choosing a door at random. Here the family has been chosen at random from the pool of families with two children, but that's totally unclear.
The annoying thing is this sits with my teaching fine, it is more my intuition that is failing to withstand trying to break it. :(
So, in the original: "a family has two children. You're told at least one of them is a girl." What are the possible states? Well, assume first born is the girl, then you have 50% that the next is a girl. Then, assume that the first born was a boy, then there is no chance and the second born is the girl that you know of. So, at 50/50 on those chances, you have 50% chance of having a 50% chance, or a 50% chance of it being 0. I can't see how to combine those to get 1/3. :(
And the Monty Hall explicitly covers the case that a decision is made on which door is shown to you. I don't see any similar framing to this problem. Yes, the total states are GB, BG, GG, but only if you treat GG in such a way that either BG or GB was not a possible state. (That is, using G for girl that you know of, and g for unknown, then possible states are GB, Gg, gG, BG. There is no version of Bg or gB that is possible, so to treat those as equal strikes me as problematic.)
Where you're getting confused is by trying to combine state space determination and probability determination at the same time (this is also why the problem is so similar to Monty Hall). The state space is shifting when you say "assume X, then the probability of Y." You are going back and forth between using and not using the information to decide arbitrarily that some probabilities are 50% and others are 0%, which leads to an invalid conclusion.
Specifically: it is not true that the firstborn has a 50-50 chance of being a girl, given you were told that the family has at least one girl. The firstborn has a 2/3rds chance of being a girl. This is the heart of your confusion.
In a broader sense there is an entire class of confusing conditional probability problems like this. Events which are causally independent in reality (e.g. gender of a child, which door Monty Hall hid the car behind) fail to be probabilistically independent when you have extra information. Yet these probability games are contrived in a way that our intuition takes over and we use our causal understanding even when a better probabilistic understanding gives you a better answer.
But in the monty hall, the "discloser" is limited by my choice/observation. It is literally part of the framing. In this framing, there is no limit based on my choice.
Consider, if you tell me that the Smiths have at least 1 girl, and I meet their daughter but you haven't met either, I have no way of knowing if I met the 1 girl or not. I could ask you, but you would just say, "I don't know, could be her. Could be the other kid. I just know they have at least 1 girl."
This is very different from the monty hall case, where the announcer knows what is behind all doors.
Similar. But different. I was using coins as an example elsewhere. If I flip a dime and a quarter, and tell you that one of them is heads, do you have increased chance of knowing if either particular one is heads? This is more liar's dice than it is monty hall.
I have no idea what point you're trying to make. I am aware the selection process is different. If you understand the argument now but you're just splitting hairs about whether it's similar to Monty Hall, fine let's agree to disagree. If you don't understand the argument and are trying to use the dissimilarity to Monty Hall as a counterargument then I don't think I can help you any further.
This is not coherent as written:
If I flip a dime and a quarter, and tell you that one of them is heads, do you have increased chance of knowing if either particular one is heads? This is more liar's dice than it is monty hall.
"Increased chance of knowing" is nonsense. What you mean is "increased chance of being correct if I guess the dime came up heads instead of tails" and this is obviously true. Given at least one of the two coins came up as heads, the probability that the dime is heads is 2/3rds, not 1/2.
Apologies, was away from computer for the extended weekend.
The crux of my view on this problem is that I can make an "at least" or an "at most" statement only knowing what one of the coins is, specifically. So, we both flip a coin, I look at mine and say "at least one is heads." You do not get an increased odds of knowing your coin, obviously. (This is the general play of liar's dice. Just, with dice. :) )
That all said, I largely land on agreeing with one of the other commenters here. The assumptions you can bring to this scenario are rather large here. The assumption I would hold would be to mirror this with how it could happen "in the world." And most ways of selecting a family where "at least 1 is a girl" has you encountering a girl. That is, ordering the two events and revealing one at a time.
In modeling this, you could say that you have two coins, one marked with heads on each side and the other a standard fair coin. In this, I can easily see how your odds of seeing heads on the first coin is increased, but odds on them being both heads is now back to 1/2. I can see how this is not the same modeling you are using, but it is still one that I have a hard time shaking from my intuition on the problem.
> 4) This extra information changes your probability estimate because the possibility of two boys has been ruled out; the naive 1/4 estimate is refined to 1/3.
That’s not correct in general.
It’s only correct if you assume that “3) Someone tells you that the family you drew has at least one girl.” was equally likely to happen whether or not there were two girls.
That’s a quite strong assumption.
One can make different assumptions and get answers different from 1/3. For example, 1/2.
Unlike the other misreading, I think you are splitting hairs about something boring and irritating. Rephrase the problem instead to "you have a reliable device that can tell whether the family has at least one daughter but doesn't tell you how many."
I understand that you may find irritating that someone points out that the original problem is ill-posed and “the answer” depends on how we decide to “rephrase” it.
However, it doesn’t seem boring in the context of a discussion of how the problem is not well-posed and additional assumptions are required to get an answer.
Really fun! Could use a hint function though. I got %100 of all words on Aug 4 but still have not found a way to get it down to zero letters. If anyone has figured it out please let me know!
I’m stuck on that one too! Got 100% of all possible words but cannot figure out how to reduce to zero. A “show hint” feature would be nice when you’re stuck.
I agree with your sentiment that animals and maybe even plants can experience sentience. But that’s a far, far, faaaar cry from what ChatGPT is and is capable of. We don’t have “general” AI yet. ChatGPT’s intelligence is very narrow; it generates written content. That’s about it. I believe a lot of people are overhyping it this way.
Bing and ChatGPT are both narrow AIs. They don’t have feelings or any real understanding of self. They simply generate text. You can ask it about itself, but anything it generates doesn’t have any real meaning, as it’s just making it up semi-randomly.
Probably the communication devices were blown off by the shockwave in the case of an implosion. It’s absolutely possible to have communications with the surface at this level; Trieste from the bottom of Mariana Trench was able to communicate back up to surface. That was 3x deeper and the early 60s.
My thoughts exactly. They should be doing a deep inspection of the hull before and after every dive. RTM sounds good until you have a failure at 10000 ft down and die instantly. “We detected a failu -“ and then you go boom.
So much skimming here eh. The whole trip can take 8 days, from launching on a ship to take you out to sea. The actual dive itself is supposed to be 12 hrs from “bolts on” pre dive to “bolts off” getting out of the sub. 8 hrs underwater, not 8 days.
Seems like they’re going to get sued if the passengers and crew aren’t saved. Possibly even if they are saved. Maybe the contract they get people to sign is tight enough, but everything I’ve read so far seems like they were a bit reckless. They don’t have to inspect their hull for issues every dive; they have real time monitoring of the health at all times! We just pop them back up when we discover an issue…
But what if that issue is detected at a depth at which nothing can be done? At a depth where death is imminent no matter what? The thing is held together with RV lights and a crappy Logitech controller from Amazon. What else have they elected to skimp on?
The “why Titan isn’t classed” page is hilarious too. They mention of the 25 pre checks they have, the majority are for surface operations, not underwater operations. Does not give one confidence and reads like a joke after this debacle.
- boy - boy
- boy - girl
- girl - girl
So it must be 1/3 chance. If you’re looking at permutations in order, that’s a different question.