Put this another way, since you say that the order doesn’t matter.
I flip a coin and hide it, asking you to guess the result.
While you are pondering, I flip another coin, and show you that it came up heads. Does that in any way affect your decision on the first one?
Or, say I have two coins that I flipped on the table, covered. I uncover one, and it is heads. Does that change the chances of the result of the other one?
These scenarios are very different than the question listed above.
No, because you are treating the two coin flips independently. You are asking me to determine the result of a single coin flip. Information about a second coin flip is irrelevant.
The riddle involves two coin flips, and you get information about the set. Saying “one of the results is heads” doesn’t allow you to differentiate between the two. It could be coin A or coin B, and since the question is about the set results it is critical information. If the question was “the first coin was heads, what are the odds of the second coin being heads” then the answer becomes 50/50.
Not sure there’s enough information here to give a clear answer. You need additional information about how many children and specific wording of the information. Depending on how the question is worded it certainly could have an impact on the answer.
1/14.
Let’s start with a girl born on mon, she has 14 possible sibs: Boy m, boy t, boy w, boy thu, boy f, boy s, boy su, girl m, girl t, girl w, girl thu, girl f, girl s, girl sun.
There are 7 such groups for a total of 98.
Of these 98, 7 are girl thu, girl x,
For a total of 7/98 or 1/14.
Unless birth order doesn’t matter, in which case there’s another 6 pairs making it
13/98
I think we have to go back to Biotop’s analysis of this. When there’s a 50/50 chance of a result — either a child is a boy OR a child is a girl; there are no other options — and we look at two separate instances of this 50/50 happening, then we get the four possibilities he outlines:
Since we have been told that ONE of the two children is a girl he is correct in eliminating possibility #1 from the mix. This now leaves us with THREE possibilities, and in only one of those possibilities is the second child also a girl. So the correct answer is that there is a 1 out of 3 (or 33.3…%) probability that both children are girls.
The fact that one of the two children was a girl who was born on a Thursday has no bearing whatsoever on the matter. It is nothing but a red herring.
-“BB”-
Birth order (or days of the week) have no relevance to gender.
Setting aside intersexed children, there are three possibilities with two children; two boys, two girls, or a boy and a girl.
One child is a girl; that eliminates the first possibility. So it’s either a boy and a girl or two girls; the chances of each are equal. The answer is 50%.
There is no missing dollar or third word ending in gry.
Sorry, that should have been TELEMARK’s analysis of this (in post #23). The rest of this has been flogging a dead horse.
https://media.tenor.com/images/f315837da880501dc7ea00982eaebb47/tenor.gif
-“BB”-
This isn’t true. It’s a way of differentiating between the two siblings, and therefore adds more information. What it enables you to eliminate more possibilities and refine the odds. In fact, the more information you get about the child the close to 1/2 the odds become.
This also isn’t true. While the two remaining options are two girls or boy and girl, the chances of those happening aren’t the same. It’s twice as likely that there will be a boy and a girl than there will be two girls.
Just as the information about the first child is irrelevant to determining the other.
Look at it this way, if I ask a large group of people with two children to raise their hands if the have at least one daughter, I have removed 1/4 of the population. What is left at this point follows a 1/3 2/3 distribution.
However, if you simply tell me that one of your children is a girl, that doesn’t give me any information about the other. You could have told me it was a boy, and it would be exactly as informative.
This is exactly like my coin flip. If I flip two coins, and cover one, what does the revealed coin say about the hidden one? Nothing. It is only if we have predetermined that I will show you a heads that it gives any information about the set.
As far as order goes, if I reveal that one coin is heads, you don’t know if it is the first or the second. So it is 50/50 there. You pointed out that order matters in HT and TH, but it also matters on HH and HH.
So knowing that one is heads, but not if it is the first or second, you have the possibilities of:
H(Hidden)H(revealed)=25%
H(revealed)H(Hidden)=25%
H(revealed)T(Hidden)=25%
T(Hidden)=H(revealed)=25%
TT=0%
So, you end up with
HH=50%
HT=25%
TH=25%
TT=0%
So, there is a 50% chance that the hidden coin is heads or tails.
Now, if a bunch of people do this, and I only go around to those who have a Heads revealed, then they do have a 2/3 chance of having the other coin be tails, but that is, as I said, only because we have already chosen a set to eliminate.
It’s kinda a reverse Monty Hall problem. If Monty Hall just opened a door at random, it wouldn’t give you any information. But since you know that he chose the door that didn’t have the prize behind it, that gives you information.
There is no difference between me pointing at my child, and asking you what gender you think it is, and then telling you that I also have a daughter, vs telling you I have a daughter and asking you if you can tell the gender of my other child.
The only way the wording matters is if we first seek out the sets that we are looking for, and eliminate the sets that we are not. In the original wording of the OP, it does not do so. It just presents us with a set with two independent variables with one revealed.
You cite agrees with me…
I tried to cut and paste, but that doesn’t work well with math off of it, but it does say that it only works out that way if we specifically look for people who qualify for the desired set, not if we actually pick a random sample.
Here’s this:
So, you have to make the assumption that he only would say that he had daughter if he had at least one daughter, and would have stayed silent otherwise. Same with her being born on Thursday. Those are unreasonable assumptions to make, unless you have intentionally sought out this set.
ETA: the original question was about a boy, the OP is about a girl, I’m talking girl in this post.
I think this is the crux of the problem. I think based on the problem description this is a reasonable assumption, so our disagreement isn’t really about the probabilities.
nm…
I suppose, if you make that assumption, but I don’t really see it being a reasonable assumption to make that a random person would only tell me about their children if they have exactly two, and that one of them was a girl born on a Thursday.
(Now, that’s an interesting probability question, if people only tell me about their kids if they have exactly two, and one is a girl born on a Thursday, how many fewer stories about other people’s kids would I have to endure?)
It really seems as though this is a set that I would have to seek out in order for the information to have any bearing on the set.
As worded in the OP, this is just some random stranger telling me about their kids, I mean that is literally what it is. I didn’t seek him out, I didn’t tell him to only tell me about his kids if they already matched certain criteria. He even used a different combination of gender and day than the original formulation, randomly chosen.
Can this be generalized? If a random person walks up to you, and tells you they have two kids and they give the gender and day of birth of one of their children, can you use that information to tell the probability of the other one? No, you can’t. It only works if they meet predetermined qualifications for the set.
It would be exactly the same as me telling you that I flipped a coin, and it came up Tails last Monday, then ask you what is the probability that if I flip (or have already flipped) another coin, it is also Tails?
The chances of a randomly thrown dart being in the center of a bullseye are significantly different if I draw the bullseye before or after I throw it. This is a case of drawing the bullseye after.
Okay, I finally get this. Even saying it s a riddle does not mean that the answer is definately going to be 13/27 because it is unknown how far I want the riddle to be pursued. It is a more complex issue than I had presumed.
How should the riddle be proposed?
BTW, in real life I have no children.
Have someone else predetemine the information. Eg.
I bumped into an old friend I hadn’t seen in years, and we started catching up. I told him I have two kids now. He laughed and said maybe I can give him some advice, since his wife had a baby girl just last Thursday. I said “that’s a coincidence, I also have a girl born on a Thursday”.
What’s the probability both my children are girls?
I think that works.
That sort of implies only one of the kids was born on a Thursday…or at least not two daughters both born on different Thursdays.
This is so brilliant that I just have to say so right away (rather than my usual practice of reading to the end of the thread to check if anyone else has already commented). Very well played indeed!
I don’t understand why you are mentioning a boy being born on a Thursday here; that doesn’t seem relevant? Ah, wait - I think you just swapped girl and boy in this paragraph.
Could the OP confirm if this (and the identical solution previously posted by Telemark in post 37) is the one they originally had in mind?
Apologies both for the triple post and if I missed it, but for those who agree 13/27 is the correct solution to the OP, how do we approach this one?
ETA: based on Telemark’s earlier statement that the more information you have, the closer to 50% the answer becomes, is it “as near to 50% as makes no difference”?
You’re going to have trouble coming up with a naturalistic format for the riddle that doesn’t have that problem. This is probably why the results seem counter-intuitive, because it only happens when you get the information via an unusual/unnatural process.
How about this:
An eccentric millionaire decided to leave her money to a random family that had a girl born on the same day of the week as her - a Thursday. Many families applied, but mine was lucky enough to win the draw. My family has two children. What are the chances that both are girls?
Not sure if it matters whether the competition is restricted to families with two children, but you could add that in if so. You can come up with your own scenario, the important thing is that something external sets the criteria, rather than you randomly choosing what to tell us.
