It’s so counter-intuitive my head is going to explode.
Here’s the paradox for the uninitiated:If I say, “I have 2 kids, at least one of which is a girl.” What is the probability that my other kid is a girl? The answer is 33.33%.
>*Intuitively, most of us would think the answer is 50%. But it isn’t. I implore you to read more about the problem.*
Then, if I say, “I have 2 kids, at least one of which is a girl, whose name is Julie.” What is the probability that my other kid is a girl? The answer is 50%.
>*The bewildering thing is the elephant in the room. Obviously. How does giving her a name change the probability?*
>
>*Apparently, if I said, “I have 2 kids, at least one of which is a girl, whose name is …” The probability that the other kid is a girl* ***IS STILL 33.33%.*** *Until the name is uttered, the probability remains 33.33%. Mind-boggling.*
And now, if I say, “I have 2 kids, at least one of which is a girl, who was born on Tuesday.” What is the probability that my other kid is a girl? The answer is 13/27.
>*I give up.*
Can someone explain this brain-melting paradox to me, please?
In: 4
This is a good example of how language and mathematical calculations can be twisted to provide you with whatever answers you want.
There are four possibilities for those children: BB, GG, BG, GB (B for boy and G for girl respectively). So if you say that one of them is a girl, you’ve eliminated the BB point, and you have a 1/3 chance that both children are girls. That’s pretty simple.
But that is only true if you cared about the ordering of the children in the first place.
The second part is just, pardon my language, stupid. First a little aside. The probability we talk about doesn’t actually describe reality, it just describes how certain we are about things. Saying a name won’t suddenly change the child from a boy to a girl, even if the probability suddenly jumped to hundred percent (look up the bayesian calculation of the likelihood that the sun will rise tomorrow if you want another fun mind twister).
But more importantly, giving the girl a name doesn’t give you any relevant information. Julie could still be either the girl in the BG pair, or the GB pair, or one of the girls in the GG pair. However, if you said “I have two children, and the older one is a girl”, that does give you relevant information. The only valid pairs are now GB and GG, so the probability that they have two girls is 1/2. In fact, this would work with any ordering, such as taller/shorter.
It’s like if I said I have a six sided die, and asked you what the probability of rolling a six would be, you’d say 1/6. If I then said “ok, but one of the faces is red”, how would that change the probability? It wouldn’t, it’s not relevant. But if I said, “actually, two of the faces show six”, well that’s different, the probability of rolling six is now 2/6.
You have to make a subtle hidden assumption to get the answer of 1/3, which is partly why the situation appears paradoxical.
Suppose you kept approaching random (honest) people and asked them “do you have exactly two children?” If they answer “no” you let them go. If they answer “yes” you ask them “is at least one a girl?”. Now if they answer “no” to that question, you know they have 2 boys, which a priori had probability 1/4. If they answer “yes”, you know they either have two girls, which has a priori probability of 1/4, or one of each, which has a priori probability of 1/2. The ratios of these scenarios must stay the same (1:2), so the probability that they have 2 girls is indeed 1/3.
Now consider this slightly different situation: your first question is the same as above. But your second question to those who have 2 children is “complete this sentence with either boy or girl so as to make it true:’at least one of my children is a …’”. Now you have two groups: all the people who complete the sentence with ‘girl’, and all those who said ‘boy’. And assuming no bias in how people answer, those groups should be the same size.
Now the first group – those who said – girl, are all people who have two children at least one of which is a girl. But the probability that the other is a girl is 50%. Because now, half the people who have both will have said ‘girl’, but the other half will have said ‘boy’.
So in the original problem, to get 1/3, we need to make an assumption as to why they said ‘girl’ rather than ‘boy’. I.e. we need to assume that they will always tell us about the girl if they have one of each. And this is, when you think about it, rather an odd assumption to make.
This is related to the Monty Hall problem, and also to the question of restricted choice in games like bridge. Information can not be considered in isolation; you also need to consider the source of the information I.e. why you received that particular bit of information rather than another. And when that isn’t random, intuitive probabilities will tend to be wrong. Eg in the Monty Hall problem, he doesn’t open a random door, he opens one he knows doesn’t contain the prize. If he opened a random one and it turned out not to contain a prize then the intuitive answer that it is 50/50 whether to swap or not would be correct.
Why does birthing order factor into the paradox?
B/G and G/B are both saying the same thing.
Does this actually track? If I asked random parents with 2 kids and have 1 girl to fill out surveys would the probability be 33% or is this a statement about a flaw in statistics
I completely disagree with the claim of 33.33% in the first place.
Consider the gambler’s fallacy, which we know is wrong, which goes like this:
“I flipped this coin and got 2 heads in a row therefore my next flip is more likely to be a tail because 3 heads in a row is only 1/8 likely. I’ve got at 7/8 chance for a tail now.”
That is the fallacy of treating outcomes that have already been set in stone (in the past) as if they were still open possibilities that are part of the probability prediction about the future.
The gambler in the fallacy is thinking of these 8 outcomes:
– HHH
– HHt
– HtH
– Htt
– tHH
– tHt
– tTH
– TTT
And forgetting that we already eliminated 6 of them by setting the first two flips in stone as both heads, so the remaining possibilities are just these:
– HHH
– HHt
– ~~HtH~~ ruled out already
– ~~Htt~~ ruled out already
– ~~tHH~~ ruled out already
– ~~tHt~~ ruled out already
– ~~tTH~~ ruled out already
– ~~TTT~~ ruled out already
Thus the chance on the next flip is now the same as it was on any normal flip. It’s 50% between the remaining 2 options.
And that fallacy is exactly what the answer “33.33%” is incorrectly doing here. It starts with only telling you there are 2 kids, which means these 4 possible open ‘future’ outcomes, like so: (“Future” as in you don’t know the information yet.)
– BB
– BG
– GB
– GG
Then it tells you it picked one and identified it as a girl, then *pretends* that only eliminated the BB outcome leaving 3 left, but it didn’t. It eliminated BOTH BB and BG because by assigning one of them making it no longer unknown, it essentially put it “in the past” in the gambler’s fallacy. It’s already been set in stone. That *creates* an implicit ordering between the two kids, in the sense that one was identified in the past, a few seconds ago, and the other will be identified in the future, not yet. And you must never use the events already set in stone in the past as if they were part of a probability calculation.
By pretending that both BG and GB are still possible, the “33.33%” answer is violating that rule. They are not both still possible. You pointed at one of the kids and said “that one is not a boy”. That already happened. That’s already a past event. So it looks like this:
– ~~BB~~ ruled out
– ~~BG~~ ruled out
– GB
– GG
The notion that this ordering that differentiates BG from GB ONLY happens if you named the one you pick is wrong. The fact that one kid has already had their sex disclosed while the other has not *IS THE ORDERING* that lets you call the already-disclosed one the lefthand letter and the not-yet-disclosed one the righthand letter, of the pair of letters.
Here are all the possible combinations of Boy and Girl for your two children:
* BB
* BG
* GB
* GG
Now we know that at least 1 is a girl, so let’s narrow that down to only combinations with at least one girl:
* BG
* GG
* GB
Out of those 3 combinations, 2 end up with the other child being a boy, and 1 with the other child being a girl. So there’s a 1/3 possibility, or 33.3…% that the other child is a girl.
>courtesy of u/Tylendal
When you flip two coins, there’s a 25% chance of TT, a 25% chance of HH, and a 50% chance of TH or HT… which is why they’re considered distinct results.
I flipped two coins thirty times and here are the results;
8 flips were both heads (27%)
7 flips were both tails (23%)
15 flips were one of each (50%)
If we consider heads to be boys and tails to be girls, then a boy/girl combo is twice as likely as a combo where the genders match.
With the gender of one of them known it eliminates 25% of the possible combinations, leaving a mixed combo twice as likely as the remaining same gender combo.
Since mixed gender combinations are twice as probable as same gender combinations that leaves a 2:3 chance that the other child is a boy and a 1:3 chance that it is a girl.
Okay, so maybe this isn’t “Like I’m 5” buuuuuut I wrote a python script to convince myself of this, in particular the part about the names. It made sense to me when I defined and ordered the filter functions. Also, could not think of female names that started with some letters…
“`
import random
random.seed()
GENDERS = [“Boy”, “Girl”]
BOY_NAMES = [“John”, “Paul”, “George”, “Ringo”]
GIRL_NAMES = [“Beyonce”, “Kelly”, “Michelle”, “LaTavia”, “LeToya”, “Farrah”, “Alice”, “Betty”, “Catherine”
, “Doris”, “Ethel”, “Francis”, “Gretchen”, “Helen”, “Izzy”, “Janice”, “Karen”, “Lauren”, “Maureen”
, “Nancy”, “Ophelia”, “Patty”, “Quigly”, “Regina”, “Stacy”, “Terry”, “Ultrabiglady”, “Vera”
, “Wendy”, “Xavia”, “Yoonis”, “Zazzy”]
class Child:
def __init__(self):
self.gender = random.choice(GENDERS)
names = GIRL_NAMES if self.gender == “Girl” else BOY_NAMES
self.name = random.choice(names)
def print(self):
print(self.name, f”({self.gender})”)
class ChildPair:
def __init__(self):
self.first = Child()
self.second = Child()
#filter functions
hasOneGirl = lambda x : x.first.gender == “Girl” or x.second.gender == “Girl”
hasTwoGirls = lambda x : x.first.gender == “Girl” and x.second.gender == “Girl”
hasBeyonce = lambda x : x.first.name == “Beyonce” or x.second.name == “Beyonce”
pairs = [ChildPair() for i in range(0, 100000)]
oneGirl = list(filter(hasOneGirl, pairs))
twoGirls = list(filter(hasTwoGirls, pairs))
oneGirlAndBeyonce = list(filter(hasBeyonce, oneGirl))
twoGirlsAndBeyonce = list(filter(hasBeyonce, twoGirls))
print(“Families with one girl”, len(oneGirl))
print(“Families with two girls”, len(twoGirls))
print(“Odds second child is girl: “, “%.3f” % (len(twoGirls) * 100.0 / len(oneGirl)))
print(“Families with at least one girl named beyonce”, len(oneGirlAndBeyonce))
print(“Families with two girls and one is named beyonce”, len(twoGirlsAndBeyonce))
print(“Odds second child is girl: “, “%.3f” % (len(twoGirlsAndBeyonce) * 100.0 / len(oneGirlAndBeyonce)))
“`
Results:
“`
Families with one girl 74828
Families with two girls 25100
Odds second child is girl: 33.544
Families with at least one girl named beyonce 3092
Families with two girls and one is named beyonce 1567
Odds second child is girl: 50.679
“`
How is everyone in this thread so wrong? The answer to the first question is 50%, not 33%. This is not a paradox in any way. The probability of the sex of one kid is entirely independent on the sex of another. A family could have 99 girls and the probability of the 100th kid being a girl is still 50%.
Everyone here seems to be breaking this up into possible family combinations, yet they overlook two of the possibilities involving the boy/boy and girl/girl combinations. So there’s B/G, G/B, B/b, b/B, G/g, and g/G. Since one is a girl, elimination the two boy combinations and we’re left with: B/G, G/B, G/g, and g/G. There’s 2/4 combinations with a second girl. The answer is 50%.
Everyone claiming otherwise is wrong. Flat out. ORDER DOES NOT MATTER
The way this is worded, it isn’t 33.33%. There’s no argument for it. When you say one child is a girl, you lock in one gender. The ordering of children is irrelevant. The only possible combos are GG and GB, because the first child, the one we know the gender of, is G.
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