Posted on 06/29/2010 4:35:59 AM PDT by mattstat
One child is a son.
The other child is either a son or a daughter.
Assume for the sake of this discussion that the birth of boys and girls is of equal probability.
The probability that the second child is a son is 1/2.
Variations on the question such as “I have two children, one a son born on Tuesday; what is the probability that my firstborn is a boy?” have different answers.
The answer to the question as posed is “1/2”.
Well, he would be right if that were what he is saying, but it is not. He is saying that the choice count is BB, BG, and GB. He is wrong. See my earlier post. I expect a retraction by Briggs.
If THAT is the question you want to ask, then in the name of all that's holy, ask it, not some other question with an entirely different answer.
My god! This is one of the most egregious examples of equivocation I have ever seen. Shame on you and on Briggs.
What is the probability that you have two boys?
Well you told us one is a boy already.
And that makes the question ambiguous. You could be asking whether given that we know one is a boy, what’s the chance the other is a boy too. That would be around 50%.
Alternatively you might be be asking is out of all possible combinations of two children, what’s the chance that you have two boys. There are four possibilities for two children. Boy-boy, boy-girl, girl-boy, girl-girl. We can eliminate, for your case, only girl-girl and girl-boy combination from consideration, since we have inside information, so to speak — we know the first is a boy. Thus only one of four combinations of two children, boy-girl, means you do not have two boys. Thus, the insider line is 75%. The inside information advantage!
But of the two ways the ambiguous question can be answered, which is it?
However you added a further specification — that 1/2 is not the answer. So now we are fairly certain of the answer — 75%!
Without looking, I think the odds are slightly better than 50%, on account of some fraction of pregnancies being maternal twins.
Just to trow some sand in the gears. ;)
One of my pet peeves is when people, discussing genetics, say “50% of your children will carry the trait”. People actually think that if they had one child without the trait, the next child will 100% have the trait. As silly as saying “50% of your children will be boys”. No. There is a 50% probability that any child will be a boy; how many of of here actually came from a family where the gender of the siblings was 50/50? Impossible for odd number of siblings, and only half or less of the even numbers.
Okay, even the answer given in the article doesn’t get it right. If we are going to handle exactitude of language, then the set of possibilities are not just boy-girl. They are boy, girl, sexually ambiguous.
And what about changes in calendar system?
Calendars have changed in history. What of some week of more or less than seven days? What’s the allowance for the annual leap-second or so? If you want to be friggin exact where do you stop?
One girl, one boy.
If you ignore twins it will be 729/1459 or 49.9657%. Very close to 50% because the odds of having both born on 29 June is very low.
I don't think ignoring twins is valid in this case because you are far more likely to have twins (I saw about 3%, but I don't know if that is 3% of births or pregnancies) than just randomly sharing birthdays (0.28% chance).
I read most of it. Now my head hurts.
But I think he’s dead wrong.
If you have one boy, the probability of having two boys after your next baby is born is 1/2.
“Boy, girl” and “girl, boy” are just two different names for the same possibility. He thinks they are two different possibilities.
I read most of it. Now my head hurts.
But I think he’s dead wrong.
If you have one boy, the probability of having two boys after your next baby is born is 1/2.
“Boy, girl” and “girl, boy” are just two different names for the same possibility. He thinks they are two different possibilities.
That sounds like the warning from my freshman physics professor in college. He said that "you will be able to solve problems in this class that you will no longer be able to solve by the time you are a senior". He meant that freshmen will happily solve d=1/2 at^2 for dropping items, while the seniors will worry about air resistance, varying gravity based on height, local variations in the gravity field from nearby mountains, friction of the dropping mechanism before the release is complete, Coriolis effect, time dilation from accelerating in a gravity field, loss of mass from radioactive decay of the dropped object, etc.
There is the problem. The wording of the tradition problem (ignoring the new Tuesday twist to it) "If I have two children and one of them is a boy, what is the probability that both of them are boys?" The way you stated it is "If I have two children and the first one is a boy, what is the probability that both are boys?"
If the four ordered options are BB, BG, GB, and GG, then the original problem throws out GG leaving BB, BG and GB equally likely, thus a 1/3 chance of two boys. The way you stated it throws out both GG and GB because you say the first one is a boy. This leaves the equally likely choices of BB and BG, so you have a 1/2 chance of two boys. Two different questions give two different answers.
The first child -- 1 out of two, or 1/2
The second child -- same as above.
Multiply the two and you get 1/4
For the Tuesday issue, multiply the .25 by 1/7, and you get your answer for at least one being born on a Tuesday.
Here we have an example of why in only score so high on IQ and such tests. I get frustrated at having to take the stupid mental lines of the idiot savants who write them. That is, at some point the difference between a 156 IQ and a 136 IQ is how inane one can tolerate being.
Where’s the good Zeno when we need him in these ‘tests’, or even a Heisenberg and his uncertainty? English is a language that only suffers exactitude in word problems so far, and this one went beyond that line of diminishing returns.
Although the reasoning, given the specified overly-nuanced parsing, once that parsing is given, is pretty interesting, and even educational.
There is a second condition: you already know that one of the kids is a boy.
The question is not "what is the probability of having a boy," it's "what is the probability of having two boys, given that one is already known to be a boy.
Mr. Bayes said much on the topic of conditional probability....
Yes, the trick in the question is realizing what it is really asking, in spite of the red herrings. I didn’t invent the question, I merely answered it correctly.
Sorry, but your answer is wrong to the question asked, for the reasons I have set out. There are not three choices to count, only 2.
You are forcing a probability into a certainty, and thereby reducing the possible outcomes. The eldest AND the youngest could both be boys, and when they are both boys, you are presuming you know which boy is being referred to, thereby exaggerating the possibility that they are both boys.
Disclaimer: Opinions posted on Free Republic are those of the individual posters and do not necessarily represent the opinion of Free Republic or its management. All materials posted herein are protected by copyright law and the exemption for fair use of copyrighted works.