# Family with a daugter/odds they have a son **URL:** \ **Category:** Cecil's Columns/Staff Reports\ **Created:** [September 12, 2013, 1:44am UTC](https://boards.straightdope.com/t/family-with-a-daugter-odds-they-have-a-son/668626 "2013-09-12T01:44:12Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![John\_W.Kennedy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_w.kennedy/32/1031_2.png) [@John\_W.Kennedy](https://boards.straightdope.com/u/John_W.Kennedy)\ **Post date:** [September 12, 2013, 8:18pm UTC](https://boards.straightdope.com/t/family-with-a-daugter-odds-they-have-a-son/668626/21 "2013-09-12T20:18:43Z")
Order doesn’t actually matter. And you don’t need to use playing cards. Just toss two coins 100 times and eliminate all the cases (about 25) with two heads. Of the about 75 remaining, about 25 will be two tails and about 50 will be one head and one tail. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [September 12, 2013, 8:20pm UTC](https://boards.straightdope.com/t/family-with-a-daugter-odds-they-have-a-son/668626/22 "2013-09-12T20:20:23Z")
> [@Chronos](#): > > Yes, exactly. > > Consider this scenario: You’re in a room with hundreds of couples, each of them parents to two children. You say “Everyone in the room whose first child is a daughter, raise your hand”, then you pick one of the couples with their hand raised. If you then ask one of those couples “What is your first child’s name?”, they’ll be able to tell you (and maybe the name they’ll tell you is Claire). Of these families, half will have two girls, and half will have a boy and a girl. > > Now, though, suppose you’re in the same room, and you say “Everyone in the room who has at least one daughter, raise your hand”, then pick one of the couples. If you ask that couple “What is your daughter’s name?”, they might tell you, or they might say “Which one?”. To a family with two daughters, the question “What is your daughter’s name?” doesn’t make sense. Of these families, 1/3 will have two girls, and 2/3 will have a boy and a girl. I have only read to the point in the thread when this was posted. Just wanted to register that it’s a wonderful explanation. It makes concrete a distinction that can be really hard to wrap the head around! ---
**Author:** ![Amateur\_Barbarian](https://avatars.discourse-cdn.com/v4/letter/a/59ef9b/32.png) [@Amateur\_Barbarian](https://boards.straightdope.com/u/Amateur_Barbarian)\ **Post date:** [September 12, 2013, 8:42pm UTC](https://boards.straightdope.com/t/family-with-a-daugter-odds-they-have-a-son/668626/23 "2013-09-12T20:42:16Z")
Okay, I’m going to be doggedly stupid here. No matter how many times I read the “2/3” arguments, they sound like variations of the gambler’s fallacy. Each flip of a coin has a 50/50 chance; each sperm-egg dance has a 50/50 chance. Minor biological possibilities aside, no one flip/dance has any effect on any other in the series. Just because one of two children is heads, how does that affect the odds of the other child being 50/50 heads/tails? The reasoning used for 2/3 seems to be erroneously reductive. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [September 12, 2013, 9:01pm UTC](https://boards.straightdope.com/t/family-with-a-daugter-odds-they-have-a-son/668626/24 "2013-09-12T21:01:53Z")
> [@](#): > > Just because one of two children is heads, how does that affect the odds of the other child being 50/50 heads/tails? The reasoning used for 2/3 seems to be erroneously reductive. There is no “the other child”. Imagine: “Do you have at least one girl?” “Yes, I do.” “What’s your other child’s name?” If a parent with two girls is asked that, how should they answer? ---
**Author:** ![Amateur\_Barbarian](https://avatars.discourse-cdn.com/v4/letter/a/59ef9b/32.png) [@Amateur\_Barbarian](https://boards.straightdope.com/u/Amateur_Barbarian)\ **Post date:** [September 12, 2013, 9:03pm UTC](https://boards.straightdope.com/t/family-with-a-daugter-odds-they-have-a-son/668626/25 "2013-09-12T21:03:53Z")
> [@Chronos](#): > > There is no “the other child”. Imagine: > > “Do you have at least one girl?” > “Yes, I do.” > “What’s your other child’s name?” > > If a parent with two girls is asked that, how should they answer? So we’re back to semanto-babble. ---
**Author:** ![Cartoonacy](https://avatars.discourse-cdn.com/v4/letter/c/848f3c/32.png) [@Cartoonacy](https://boards.straightdope.com/u/Cartoonacy)\ **Post date:** [September 12, 2013, 9:06pm UTC](https://boards.straightdope.com/t/family-with-a-daugter-odds-they-have-a-son/668626/26 "2013-09-12T21:06:32Z")
> [@bump](#): > > 1 and 2 are equivalent. For the purposes of the question, it doesn’t matter if it’s Girl/Boy or Boy/Girl. It does matter. If a family has two children – or if you pick any two people at random, for that matter – there’s a 50% chance that they’ll both be the same sex, and a 50% chance that they’ll be different sexes. If they’re the same, that there’s a 50% chance (25% overall) that they’re both male, and 50% (again, 25% overall) that they’re both female. If you don’t believe that, then try my experiment with the cards, or John W. Kennedy’s with the coins. (I like cards better, but that’s just me.) ---
**Author:** ![Cartoonacy](https://avatars.discourse-cdn.com/v4/letter/c/848f3c/32.png) [@Cartoonacy](https://boards.straightdope.com/u/Cartoonacy)\ **Post date:** [September 12, 2013, 9:11pm UTC](https://boards.straightdope.com/t/family-with-a-daugter-odds-they-have-a-son/668626/27 "2013-09-12T21:11:10Z")
> [@Amateur\_Barbarian](#): > > Just because one of two children is heads, how does that affect the odds of the other child being 50/50 heads/tails? It doesn’t. And that’s why the answer is 2/3. You flip two coins. Each coin has a 50% chance of landing heads or tails. Both flips are independent of each other. If coin A lands heads, there’s a 50% chance that B will be heads, and 50% that it will be tails. If coin A lands tails, there’s still a 50% chance that B will be heads, and 50% that it will be tails. So, because both flips are independent, there’s a 50% chance that they’ll come up different, a 25% chance that they’ll both be heads, and a 25% chance that they’ll both be tails. Similarly, suppose you have one child. Whatever sex that child is, there’s a 50% chance that your next child will be the same sex, and 50% that it will be of the opposite sex. ---
**Author:** ![bump](https://avatars.discourse-cdn.com/v4/letter/b/7c8e57/32.png) [@bump](https://boards.straightdope.com/u/bump)\ **Post date:** [September 12, 2013, 9:21pm UTC](https://boards.straightdope.com/t/family-with-a-daugter-odds-they-have-a-son/668626/28 "2013-09-12T21:21:39Z")
Ok… real world example. I have a wife and a son. Let’s hypothetically say my wife’s currently pregnant. Are you telling me we have 66% chance of having a daughter for our second child? ---
**Author:** ![Cartoonacy](https://avatars.discourse-cdn.com/v4/letter/c/848f3c/32.png) [@Cartoonacy](https://boards.straightdope.com/u/Cartoonacy)\ **Post date:** [September 12, 2013, 9:26pm UTC](https://boards.straightdope.com/t/family-with-a-daugter-odds-they-have-a-son/668626/29 "2013-09-12T21:26:43Z")
> [@bump](#): > > Ok… real world example. I have a wife and a son. Let’s hypothetically say my wife’s currently pregnant. > > Are you telling me we have 66% chance of having a daughter for our second child? Just the opposite. You have a 50% chance of a son and a 50% chance of a daughter. As I said, there’s a 50% chance that your second child is the same sex as the first. There is, however, a 0% chance that both of your children will be daughters. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [September 12, 2013, 9:30pm UTC](https://boards.straightdope.com/t/family-with-a-daugter-odds-they-have-a-son/668626/30 "2013-09-12T21:30:34Z")
> [@bump](#): > > Ok… real world example. I have a wife and a son. Let’s hypothetically say my wife’s currently pregnant. > > Are you telling me we have 66% chance of having a daughter for our second child? Go grab a thousand couples who have a son and who are pregnant. I’ll wait. Okay, got 'em? Now go give them all an ultrasound, and tell me what percentage of the fetuses are female. About fifty percent. Okay. Now let’s do another. Go grab a thousand couples who have two kids. Again, I’ll wait. (No cheating, make it random. Just screen for “having two kids” and nothing else.) Alright, now ask them to raise a hand if they have at least one daughter. How many raised their hand? About seventy five percent, right? Okay, now dismiss the other twenty five percent. You’ve got 750 couples remaining. _Now_ ask the 750 to raise their hand only if they have a son. How many raised their hand? About 500–67 percent of them. There it is. Are you doubting that any of the above would actually be true were the scenarios actually carried out? ---
**Author:** ![Amateur\_Barbarian](https://avatars.discourse-cdn.com/v4/letter/a/59ef9b/32.png) [@Amateur\_Barbarian](https://boards.straightdope.com/u/Amateur_Barbarian)\ **Post date:** [September 12, 2013, 9:31pm UTC](https://boards.straightdope.com/t/family-with-a-daugter-odds-they-have-a-son/668626/31 "2013-09-12T21:31:12Z")
> [@Cartoonacy](#): > > JThere is, however, a 0% chance that both of your children will be daughters. I don’t get it. 🙂 ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [September 12, 2013, 9:34pm UTC](https://boards.straightdope.com/t/family-with-a-daugter-odds-they-have-a-son/668626/32 "2013-09-12T21:34:20Z")
nvm ---
**Author:** ![Heart\_of\_Dorkness](https://avatars.discourse-cdn.com/v4/letter/h/4bbf92/32.png) [@Heart\_of\_Dorkness](https://boards.straightdope.com/u/Heart_of_Dorkness)\ **Post date:** [September 12, 2013, 9:34pm UTC](https://boards.straightdope.com/t/family-with-a-daugter-odds-they-have-a-son/668626/33 "2013-09-12T21:34:45Z")
> [@Amateur\_Barbarian](#): > > Just because one of two children is heads, how does that affect the odds of the other child being 50/50 heads/tails? It doesn’t. It _seems_ to, but it doesn’t. As you and Chronos point out, how you pose the question makes all the difference. The gambler’s fallacy is that if I’ve just thrown 19 heads in a row, the chances of me throwing heads on my next toss are really small. In fact, they’re still 50/50. _On the next toss_. But if instead of asking “What are the odds that this next toss comes up heads?” you ask, “What are the odds that I can throw 20 heads in a row?”, then you get the teeny tiny odds you’re looking for. Similarly, this question _isn’t_ asking, “If I have one daughter, what are the chances that the second child is a boy?” That would be indeed be 50/50, or 1:2. It’s asking, “Of all the families that have two kids, let’s eliminate those without at least one girl. Now, of the remaining families, what are the odds that _both_ kids are girls?” Or if you want to put it in coin flips, it’s asking, “Consider all possible two-flip scenarios. Eliminate those scenarios that have no heads. Now, of the remaining scenarios, what are the odds that you have _two_ heads?” It’s tricky because it sounds straightforward, but it’s a really weird question, when you get down to it. And that’s why it’s interesting. It sheds light on flaws in common-sense approaches to problem solving. A more standard question would be, “Consider all possible two-flip scenarios. What are the odds of getting two of a kind?” Your odds are 1:2. Obviously, right? Because whatever your first throw was, the next one will either be the same or different. So you have a 1:2 chance of getting the same thing on the next throw. Or perhaps, “Consider all possible two-flip scenarios. What are the odds of getting two heads?” Your odds are 1:4. Now, even here, some people might say, “It should be 1:3! I can get two tails, a head and a tail, or two heads!” But again, not all possibilities are equally likely. The only way to correctly calculate the odds is consider what can actually happen at each juncture and multiply it out. The first throw has a 50% chance of being heads, and so does the second. This yields four different equally likely scenarios - TT, TH, HT, and HH - each with a 25% or 1:4 chance. This means that throwing a head and a tail (in any order) is twice as likely as throwing two heads. But this question is basically saying, “Consider all possible two-flip scenarios. Half are two of a kind, and half are two different sides. Then, from the half that are two of a kind, eliminate half of _that_ group: all the two-tails. Now, you’re left with 3/4 of your original scenarios: the half that were two different sides, and the _two-heads half of_ the half that were two of a kind. Of these remaining scenarios, how many also have a tail? How many have two heads?” At a glance, it doesn’t sound like nearly as complicated an operation as it actually is. And that’s why stuff like this is interesting to me: we can make huge errors in logic because of small changes in language, and it behooves us to be on our guard for things like this that we encounter in everyday life. “Wait… what is this graph _really_ saying?” ---
**Author:** ![Cartoonacy](https://avatars.discourse-cdn.com/v4/letter/c/848f3c/32.png) [@Cartoonacy](https://boards.straightdope.com/u/Cartoonacy)\ **Post date:** [September 12, 2013, 9:35pm UTC](https://boards.straightdope.com/t/family-with-a-daugter-odds-they-have-a-son/668626/34 "2013-09-12T21:35:06Z")
> [@Amateur\_Barbarian](#): > > I don’t get it. 🙂 His example started with “Ok… real world example. I have a wife and a son.” If he specifies that one of his two children is a son, then we can eliminate the possibility that both children are daughters. For that matter, we’re also eliminating the daughter/son combination, leaving two equal possibilities: son/daughter or son/son. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [September 12, 2013, 9:35pm UTC](https://boards.straightdope.com/t/family-with-a-daugter-odds-they-have-a-son/668626/35 "2013-09-12T21:35:22Z")
> [@Amateur\_Barbarian](#): > > I don’t get it. 🙂 I’m not sure what the smiley face means here. Are you saying “I don’t get it, but I remain friendly?” Or are you saying “I don’t get it, but actually I’m just kidding and I do get it?” ---
**Author:** ![Powers](https://avatars.discourse-cdn.com/v4/letter/p/edb3f5/32.png) [@Powers](https://boards.straightdope.com/u/Powers)\ **Post date:** [September 12, 2013, 10:39pm UTC](https://boards.straightdope.com/t/family-with-a-daugter-odds-they-have-a-son/668626/36 "2013-09-12T22:39:43Z")
> [@Terr](#): > > You haven’t met any in the first statement. In that case, _that’s the difference_ between the two statements: in one, you know they’re speaking of a specific girl. In the other, they could be speaking of either of their two children. Powers &8^] ---
**Author:** ![Powers](https://avatars.discourse-cdn.com/v4/letter/p/edb3f5/32.png) [@Powers](https://boards.straightdope.com/u/Powers)\ **Post date:** [September 12, 2013, 10:43pm UTC](https://boards.straightdope.com/t/family-with-a-daugter-odds-they-have-a-son/668626/37 "2013-09-12T22:43:38Z")
> [@bump](#): > > Ok… real world example. I have a wife and a son. Let’s hypothetically say my wife’s currently pregnant. > > Are you telling me we have 66% chance of having a daughter for our second child? No, you have a 50% chance of having a daughter, because in this situation, you’ve already eliminated two of the four different ways of having two kids: A. boy, girl B. girl, boy C. girl, girl D. boy, boy Each of these is equally likely, but you’ve eliminated B and C by telling us your first child was a boy. That leaves A and D, and each has equal probability, or 50%. Powers &8^] ---
**Author:** ![bump](https://avatars.discourse-cdn.com/v4/letter/b/7c8e57/32.png) [@bump](https://boards.straightdope.com/u/bump)\ **Post date:** [September 13, 2013, 1:58am UTC](https://boards.straightdope.com/t/family-with-a-daugter-odds-they-have-a-son/668626/38 "2013-09-13T01:58:53Z")
> [@Powers](#): > > No, you have a 50% chance of having a daughter, because in this situation, you’ve already eliminated two of the four different ways of having two kids: > > A. boy, girl > B. girl, boy > C. girl, girl > D. boy, boy > > Each of these is equally likely, but you’ve eliminated B and C by telling us your first child was a boy. That leaves A and D, and each has equal probability, or 50%. > Powers &8^] 1. This isn’t the Monty Hall question, and that’s why I think applying that logic to this one is wrong. You’re right in my example that I’ve ruled out B and C, but in the OP’s example, there’s no difference between B and C because it doesn’t matter who came first. Just knowing that a couple has a daughter doesn’t change the odds that they may also have a son, assuming that sons and daughters are born with equal frequency. Explain how my question is different from the OP’s question. There seems to be no difference from a logical and practical standpoint. ---
**Author:** ![ZenBeam](https://avatars.discourse-cdn.com/v4/letter/z/3ab097/32.png) [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)\ **Post date:** [September 13, 2013, 2:05am UTC](https://boards.straightdope.com/t/family-with-a-daugter-odds-they-have-a-son/668626/39 "2013-09-13T02:05:41Z")
> [@bump](#): > > Explain how my question is different from the OP’s question. In your question, you know the sex of _one particular child_ (your son). In the OP’s question, you only know that one of the two children is a girl. You don’t know the sex of a particular child. That’s the difference. ---
**Author:** ![ZenBeam](https://avatars.discourse-cdn.com/v4/letter/z/3ab097/32.png) [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)\ **Post date:** [September 13, 2013, 12:05pm UTC](https://boards.straightdope.com/t/family-with-a-daugter-odds-they-have-a-son/668626/40 "2013-09-13T12:05:56Z")
> [@ZenBeam](#): > > Look, just make a sample of four families: > 1 Girl, Boy > 2 Boy, Girl > 3 Boy, Boy > 4 Girl, Girl > > “everyone who’s first child is a daughter”: 1 and 4 raise their hand. Half have a boy. > "Everyone who has a second child that is a daughter ": 2 and 4 raise their hand. Half have a boy > “they have at least one daughter”: 1, 2, and 4 raise their hand. 2/3rds have a boy. > [@bump](#): > > Here’s the problem with your reasoning. > > 1 and 2 are equivalent. For the purposes of the question, it doesn’t matter if it’s Girl/Boy or Boy/Girl. Then later you wrote: > [@bump](#): > > Ok… real world example. I have a wife and a son. Let’s hypothetically say my wife’s currently pregnant. So to further answer this: > [@bump](#): > > Explain how my question is different from the OP’s question. Cases 1 and 2 are in no way equivalent for you. You have zero chance of ending up with Case 1, with a girl for your first child and a boy for your second, and a 50% chance of ending up with an older boy and a younger girl. But in the OP’s question, when all we know is that at least one child a girl, Case 1 _is_ possible, and just as likely as Case 2 or Case 4. That’s how the questions are different. [Previous page](https://boards.straightdope.com/t/family-with-a-daugter-odds-they-have-a-son/668626.md?page=1) [Next page](https://boards.straightdope.com/t/family-with-a-daugter-odds-they-have-a-son/668626.md?page=3)