Boy girl paradox

Added: 2026-06-01

[00:00:00]This is boygirl paradox. What I'm going to do is give the answer uh pretty quick, show a quick proof, and then um yeah, you can stop the video after that, but if you want, then I'm going to show some more proofs and I'll show why a certain method is wrong. And that's the method that I saw every mathematician use on YouTube. All right. So, boy, girl, paradox. We have a mother with two offspring.

[00:00:26]Maybe because they're the offspring. I thought about saying that when they get to be teenagers, one is a loser with no self-esteem and they're always fighting.

[00:00:33]So, the mother has to keep them separated. Or maybe I won't say that.

[00:00:37]Okay, so two offspring, one is a boy.

[00:00:40]What is the probability both are boys?

[00:00:43]It's either 1/2 or 1/3.

[00:00:46]I'll give the answer in 3 2 1 if you want to pause it. Um, it is 1/2. If you get 1/3, it's because you're using the wrong model. And that's the model that all the mathematicians used on YouTube and the videos that I saw. All right. Uh first I want to say I grew quite a few synapses uh working on this problem.

[00:01:06]This is about my fourth video on this.

[00:01:09]The learning theory is every time you learn something you grow a synapse between a couple of neurons in your brain. So over the last several months or over the last year, every once in a while I look at this and I figure out something new or a different proof. So anyways, it looks like I'm growing a few synapses. Also, this video really uh teaches you uh to work on your soft skills of um which would be teaching or explaining or educating. The hard skills would just be uh to figure out the problem and understand it. But the soft skills such as communication is to be able to teach this. And that's that's where the art comes in. So, I grew quite a few synapses trying to figure out how to explain this clearly so that a child could understand. And I'm going to be using just elementary school math here and logic. So, this is easy stuff. It's just there's a lot of it and it took me a long time to put it together. So, you could think of it as a pretty complicated jigsaw puzzle with a 100 pieces or so. And yeah, I had to fit them all together for this video. At any rate, boy, girl paradox. I'm going to prove that one answer uh is correct and that answer is 1/2. I'm also going to prove that the other answer is incorrect and that's 1/3. I'm going to show three models. Uh the first model I show is correct, second model is incorrect and then the other model is correct. But anyways, I'll show those now. All right.

[00:02:32]So to build our models, we start with the u uh flipping a coin. On your first flip, you either get a heads or a tails.

[00:02:39]On the second flip, you either get a heads or a tails. This is equivalent or analogous to having a baby. Uh the first baby will either be boy or girl. Second one will either be boy or girl. So you can just replace the word heads with boys and tails with girls if you want.

[00:02:55]Here's all the possible combinations when you flip these coins. You can get a head head tail tail head tail tail. List them out in order right here. As you can see, since we're dealing with a double flip or we're flipping two coins, 1/4 of the time you're going to get a head head. This is a double event. If you're placing a bet on this event, you're betting on what's going to happen with uh two of these coins uh instead of just one. So, 1/4 of the time you're going to get two heads in a row. But also notice one half of the time you're going to get a head. One half of the time you're going to get a tail. And if you're making a bet or placing a bet on this, this is a single event. Count them up on the heads. One, two, three, four. Four heads. One, two, three, four. Four tails. showing that it's always 50/50 whether you get a head or a tail.

[00:03:41]Another way to look at this is you have a 1/ half probability of getting a heads if you flip heads or tails. One/2 heads.

[00:03:48]Then if you get another heads, this is going to be 1/2 of 1/2. So this head right here, uh, this head is half of this whole group. So this is 1/2 of 1/2. 1/2* 1/2 equals 1/4.

[00:04:02]So you get 1/4 head heads. If you do three flips, then it goes uh uh 1/2 of 1/2 of 1/2. You just keep doubling the denominator. So it goes 1/2, 1/4, 1/8, 1/16th. If you want to keep getting heads here, I said uh just imagine we're doing 10 to the six trials. So let's say we're doing a million trials in this thought experiment. Coins hit the table at the same time. So there is no first or second flip. Uh you just have two coins in a cup and you dump the cup on the table. So they both hit at the same time. That way we don't have to worry about um older, younger, or first or second.

[00:04:37]And so that takes care of the heads, tails. Let's do the boy, girl. Now, same thing. Boy, girl, uh sorry. If you have a birth, the first birth will be boy, girl. Second birth will be boy, girl.

[00:04:48]So, you have either boy, boy, boy, girl, girl, boy, girl, girl. Um, line them all up right here. Let's say these babies are all the same age as well, because you don't need to have older or younger.

[00:04:59]They could all come out at once with a C-section. These could be twins or triplets or whatever. Um, or it could be a litter of puppy with just two puppies in the litter. Also of note, this part right here where it says boy one and boy two, boy two is the same boy. In this case, boy two has a brother. In this case, boy two has a sister in an alternate universe. So, boy number two here, his name could be Schroinger because he exists in two places at once depending on who um comes out with him.

[00:05:30]Either he is a twin with a brother or he is a twin with a sister. And if you want you could say that he is the younger brother or sorry you could say he has an older brother or you could say he has an older sister. It doesn't matter if they happen sequentially or simultaneously.

[00:05:46]So I just said here he either has a brother or a sister. So it's important to understand these flow diagrams or this tree diagram as um as we continue to move through this. So I said here same age they could be a C-section or litter of puppies. So for your families, if you're going to place your bets on this, you got a million families. Um 10 to the six families. So a million families. They each have uh two offspring. Uh 1/4 of the families will be boy boy. Um then uh 1/4 of the family is boy girl, 1/4 girl, boy, 1/4 girl.

[00:06:19]And if these all come out as twins at the same age, then you could combine family two and family three and say that you have one half uh boy girl families.

[00:06:27]Now, if you're going to place your bets here, um 1/4 of these families will be boy boy. And also, uh 1/2 will be boys, one half will be girls. You can count them up. One, two, three, four boys, 1 2 3 four girls. Um but if you're betting on just either a boy or a girl, that is a 1/ half probability. And if you're betting on a double outcome of having a two boy family, that's going to be a 1/4 probability. I did a separate video on uh relating probability to how much you need to win on every bet to uh break even. You can see it if you want to.

[00:06:58]Next page. Here's the proof that one half works. Let's take our two coins.

[00:07:03]You flip two coins in a row or at the same time there's a 1/4 probability of head head. Boy. And that is if you know nothing about either coin. So you're you got two coins on a table. They were just dumped out in a cup. You cover those coins up with a napkin. You pull the napkin away. You have a 1/4 probability of guessing head head. The next uh situation though, we know that one is a head. What is the probability that both are heads? This is the boy girl paradox.

[00:07:30]We know that the mother has one boy.

[00:07:32]What is the probability both are boys?

[00:07:34]They use the word both. And this is what's very very confusing. This makes it look like a double event that we're betting on, but it is actually a single event. And I got proof for all this. So, one coin is a head. What's the other one? It's either going to be heads or tails. And I said here we are asking about a single outcome. So one half will be heads or boy. I said here see the tree diagram because we could be looking at the first or second head. And what this means is we have a head right here on the left.

[00:08:07]That could represent this head right here. So what can you get after this head? One coin hits the table. It's a head. The other coin could either be a head or a tail. That's 50/50. But what if this head right here is actually this head that we're looking at? This head could either be paired with a head or a tail. So you could follow the this flow diagram uh forward or backward and you have a 1/ half probability. So it is not 1/4 because 1/4 means we know nothing about either coin. But once we know something about a coin, it's yeah, it's 1/2 probability because we're betting on a single event. And I have more proof for that. First off, there's a gamblers fallacy and I'll mention that down here.

[00:08:46]The gambler's fallacy states that um a previous event cannot affect a future event. Okay, so let's test it. Let's prove it. I put a star next to it there because we want to prove it. We'll do 10 to the six trials, a million trials. The head family will be 1/4th as I've already shown. 1/4 of all flips. When you do these double flips, 1/4 of those will be head. Now, every time you get a head, record the next flip. What will the next flip be? Half the time it'll be a head, the other half the time it'll be a tail. Uh, lest you believe in the gambler's fallacy, which states that a previous outcome having a head here will affect a future outcome. And that is not true. Next page. Oh, actually, sorry.

[00:09:32]That's actually my proof. That's it.

[00:09:34]Now, I'm going to show some other proofs and show a method that is wrong. All right. So, the first model I showed was right here. I think I wrote it, but I think I forgot to say it. So, that's the two coins. That's the page I just showed. This is the first model right here. Um, just following the tree diagram. Now, I'm going to show model number two, and show why it's wrong. Got our uh pairs lined up here. We have heads, head one and two, head three, head four. You can see how they're lined up. Head, head, head, tail, tail, head, tail, tail. 1/4 of these families or these pairs will be head or 1/4 of the families will be boy, boy. Now we know that since uh one offspring is a boy then that means we cannot have a girl girl option. So we eliminate tails tails right there. No girl girl. Now we go from 1/4 probability of head head to 1/3 probability of head head. So it goes head head tail and tail head. Nothing magical. We just go from 1/4 to 1/3 because we just eliminated our sample right there. So, in other words, if you have four cards on a table and they're labeled as head, head, head, tail, tail, head, and tail tail, you have four cards. You have a one out of four probability of picking the head head card. But now, we've uh cleaned up all of the tailtail cards off the table.

[00:10:48]Now, you have a one out of three probability of picking a head head card.

[00:10:52]And that is all this model is good for.

[00:10:54]It just tells you the probability of picking a boy boy family. And that's why this is a wrong model. But I'll keep going. I just thought of that scene from The Princess Bride. You truly have a dizzing intellect. Wait till I get going. So anyway, that's what I'm doing here. I'll keep going. Okay, so continuing with the wrong model. Uh, boy, boy, boy, girl, girl, boy. One-/ird is a boy, boy, family. Okay, so this is the wrong model, but what we need to do is modify this model to what I'll call model number three. Because now we know that that mother is with a son. The mother has one son. Maybe we see them in public. We see mom in public with her son. She is either going to be with boy one, boy two, boy three or boy four. Uh, one mom will be with boy one and two.

[00:11:39]The other moms will be with boy three or boy four because we have three families.

[00:11:43]Okay. So now we have to decide uh boy one, boy two, boy three, boy four. Will they have a brother or a sister? You can just look up here. Boy one will be paired with a boy. Uh, boy two paired with a boy. And I'll show that on the next page.

[00:11:58]Okay, so boy one has a brother, boy two has a brother, boy three has a sister, boy four has a sister. Even if you eliminate the girl group from the study, there will still be a 1/ half probability of a boy with every birth.

[00:12:15]And you will have a 1/ half probability um every boy will have a brother. So here's your boys right here. one half probability of having a brother, one half probability of having a sister because there are two brothers and two sisters.

[00:12:31]So, model two is wrong, but modify it to model three and you'll get the right answer. I'll keep going. Why does model number two not predict the gender of the other person? And that is because it just tells you that if you eliminate the girl category, then you will have a one-third probability of choosing a boy boy family. So they're like cards on a table laying face down. Instead of having four cards on a table, now you only have three cards on a table. So you have an increased probability of choosing a boy boy family.

[00:13:07]And then I said here, but choose any boy and one half will have a brother and one half will have a sister. And thus showing that the Mandelian genetics still hold true. I believe this may be the cause of confusion. One-third of moms will be in a boy family, but one half of boys will be in a boy family. And that's because we did have 1/4 of the mobs that were in a boy family or 1/4 of the coin pairs were head head, but we eliminated the girl category. So it goes up from 1/4 to 1/3.

[00:13:43]And the boys, it was two4s of the boys that are in a boy family. Two boys have a brother, two boys have a sister, but that reduces to 1/2. and eliminating the girl category doesn't change that. Okay, so here's the sophistry. Every mathematician I saw on YouTube said that um if a mother has a boy, she only has a one-third probability of having another boy. Any mathematician should know better because first off, reason number one, we had the mandelian genetics in high school biology. You simply draw your opponent square xxxy.

[00:14:22]The mom's uh egg will either have an XX.

[00:14:25]I believe those are called gametes.

[00:14:26]They're hloid. They only have half the DNA. So the mom's egg will have the sex chromosome of an X. The dad's sperm, his gameamt, will either carry an X or a Y sex chromosome. If an X sperm fertilizes an X egg, you get XX. So therefore, you get a girl. If it's a Y sperm that fertilizes the X egg, you get a boy.

[00:14:48]Thus you have a one half probability of girl, one half probability of boy. There is a very slight uh variation in this or a couple of nuances. Um extremely rarely uh you could have uh more than one uh sex chromosome. So you can get ambiguous gender and also one gender is slightly more commonly born than the other but it's an extremely narrow margin. And there's a theory on that that one gender can have a slight increase in a chromosomal abnormality that could lead to u the pregnancy not being carried to full term. So that means it could either lead to a silent or a known miscarriage.

[00:15:29]But that is minutia uh pedentry isn't always bad because it really forces us to uh think and speak uh more precisely.

[00:15:38]All right. So any rate I just said one half boy, one half girl from the punet square. Um, so, um, any mathematician is going to learn that in high school.

[00:15:45]They're going to have it redundantly taught to them in bio 101 in college.

[00:15:50]So, yeah, everybody learns that twice if you go to college, and you learn it once if you go to high school. Next up, as you can see from the punet square, it is the father that determines the gender of the offspring. The father doesn't consciously determine it, but it's the father's gamet that determines it.

[00:16:07]because that a sperm cell that carries either a X or a Y sex chromosome, that sperm cell will have no idea what the um gender was of the previous baby that occupied uh that uterus. Next up, uh there is the gamblers's fallacy that every mathematician will learn. Just basic statistics that just shows or that tells that a previous outcome cannot affect a future outcome. And just for fun, a effect is an action, so it's a verb. And effect is a noun. Thus, if you kick someone, then you affect them. But the effect will be they get hurt or they get their feelings hurt. It is strange that someone like myself that just dabbles in math for fun could solve this or could solve what no mathematician on YouTube could solve, at least uh of all the videos I saw. So, this shows that I either have some sort of exceptional ability, uh, like the biologist in Big Bang Theory that wins a Nobel Prize in physics because apparently there's no difference between biology and physics, or all the math videos that showed up on my homepage uh, were just trolling. I'd like to think it was the first reason and not the