Since the constraint simply says that at least one child is a boy, we don't need to distinguish between two types of BB.
This is a curious misconception, by the way. The typical failure mode I see on these questions is people having difficulty accepting BG and GB as different possibilities.
If we are determined to look at order of birth (ie. have a BG and GB) then we should consider all cases by order of birth, so where 'B' represents the boy we know and 'b' or 'g' represents a child we dont, and the first character represents the first child, and the second the second childe - we have:
Consider all the possibilities and then look at the outcomes they produce.
BB
BG
GB
GG
These are all equally likely, right? They each have P = 1/4. If we took 1000 fathers who had two children each, and lined them up, we would expect 1/4 of them to have children matching each of the above pairings. That means 250 of each. With me so far?
Now, we know that we are dealing with a father who has at least one son. If we went along the line and asked each father 'Do you have at least one son: Yes/No?', then 750 would answer yes, and 250 (the GG fathers) would answer no.
If a random father comes up to us and says 'I have at least one son', we know that he is from those 750 - we have selected a subset to deal with. Of those 750, only 250 have a second son, so the probably is 250/750 or 1/3.
You're assuming a different selection model, in which a father of two boys is twice as likely to volunteer information if he has two boys: "Pick a child at random, then if it's a boy - say: I have a boy [blablabla] what is the other?"
However, you're doing this on an already selected sample by discounting all the girl-girl pairs. The selection rule in your logic then becomes a two-step "If you have at least one boy, then pick a child at random, then if it's a boy, say: [blablabla]"
Given that, your analysis is correct.
But, seeing as it is a wildly different selection model than what everyone else seem to work with, you should be explicit about it.
Okay. To respond to both of you - I have put this in programming just to check, and I think you are taking a different inference from the question than I am.
Under your inference, the man wouldnt have mentioned anything unless he had at least one male child. (in which case you can say the GG scenario is gone, but GB BG and BB are equally probable)
Under my inference, the man just told me the sex of one child at random.... (in which case BB is twice as probable as GB or BG - where he could have equally said 'i have at least one girl')
That sound reasonable to you? I have it in ruby form if you are interested :)
The blogpost linked to by someone above (http://blog.tanyakhovanova.com/?p=221) uses this explanation, which is very different (to me) than the one in the main link, and I can see why this gets to 1/3:
"A father of two children is picked at random. If he has two daughters he is sent home and another one picked at random until a father is found who has at least one son."
"Under your inference, the man wouldnt have mentioned anything unless he had at least one male child. (in which case you can say the GG scenario is gone, but GB BG and BB are equally probable)"
No. I'm just assuming he has two children, and randomly mentions something about one of them. The GG scenario is only eliminated after he makes his statement, because we then know he has at least one boy.
Put it this way. before he says it, we have
GG, GB, BG, BB
after he says "I have a child that is [MALE OR FEMALE]" we have (where the capital letter is the child whose sex has been mentioned, and the lowercase letter is the other child):
Gg, gG, Gb, gB, Bg, bG, Bb, bB
So if he has said the sex is male, then we have four combinations left:
gB, Bg, Bb, bB.
Understand that we go to more scenarios (8) based on which child is mentioned, before we go to fewer. Actually the order of the children is something you can and should ignore, however as you are holding on to it, I show it this way....
I'm afraid this is wrong - you shouldn't distinguish between Bb and bB. In this problem, they are not different states, so counting them messes up your probability calculation.
For a randomly selected family with two children, there are four possible boy/girl combinations:
B B,
B G,
G B,
G G
In the first case we are told that the older child is a boy. This leaves only two cases:
B B,
B G
Therefore, there is a 50% chance the second child is a boy.
In the second case, we are told only that [at least] one child is a boy. This leaves three possibilities:
B B,
B G,
G B
Therefore, the probability that both children are boys is 1/3.
Enumerating possible states of the world like this is the fundamental insight you need to have to be able to understand these types of problems - but it does take a while to get used to!
But you are assuming that each of those three possibilities has equal probability; can you explain the rationale for that? (it is clear why BB, BG, GB, and GG have equal probability in the unrestricted case, but less clear why BB, BG, and GB have equal probability in this restricted case)
Besides, this is just a rephrasing of the original article's argument, and doesn't counter mine at all. I am open to the possibility that there is a flaw in my argument, but where is it?
So you say: "it is clear why BB, BG, GB, and GG have equal probability in the unrestricted case"
The restricted case is just the unrestricted case + one additional bit of information, that is, you're told that GG is not an option. This eliminates GG from the unrestricted case, but says nothing more about the probabilities of the other options. So the probabilities stay equal, although they now equal 1/3 each (if you eliminate options, the remaining options all become more likely).
What you're missing is this: The statement "the older child is a boy" has more information than the statement "one of the children is a boy". The first statement allows you to eliminate two options (GB and GG), while the second statement only allows you to eliminate one option (GG).
The "older" part is not fundamental to the problem. Equally, the statement "the taller child is a boy" has more information than "one of the children is a boy". The problem with this is that probabilities for height are not so friendly like the 50/50 probabilities for birth order (e.g., boys are likely to be taller than girls, older children are taller than younger, etc), which introduces unnecessary complexities to a logic problem. So that's why birth order is used for these types of puzzles.
Penicillin is produced, distributed and sold. I've bought it more than once and consider it to be a product. What do you feel is the necessary criteria for something to called a product?
The funny thing is that he is clearly thinking like a computer programmer in his first sentence. I wonder if naive users would generally have better results on first contact with Siri, compared to technical users who try to over-structure their commands?
You're very possibly right. That said, the "Add ketchup to my shopping list" version only sounds good for shopping lists.
If I wanted a list of work reminders, I might have: "Add send contract to john to my work list." which would be less natural... also, in my mind, what I was adding to my reminders was the task to buy ketchup, not "ketchup" by itself.
Anyway, I'm sure they'll add more supported syntaxes, and in parallel people will learn and adapt to the syntaxes that are actually supported.
It can. As CEO, one of his responsibilities was to create an organisation that would endure. Apple itself may be Steve Job's greatest creation. The test of the next 10 years is to see if that is true.
As far as we know he didn't even organize his own succession, which is one of the most basic responsibilities of a CEO of a multinational. Even when he knew he was terminally ill. So I don't see much hope that he organized the rest of Apple as some great enduring company either.
Steve didn't make his succession plans public, but that doesn't mean they didn't exist. Publicly announcing your succession plan has no upside and lots of downside. Just ask Jay Leno or Brett Favre.
So the theory here is that the greatest CEO of all time was able to mind-meld his vision, drive, creativity, discipline, and all his other forms of genius into the corporate structure of Apple? Oh brother. Do you guys hear yourselves?
Are you saying Steve couldn't train others in what is impotent and what is not? The only choices are "Steve was the only Apple employee" or "Steve had no influence at all"?
None of the Apple execs are the same people they were 10 years ago. This is a joint interaction between them and Steve. Those changes have the potential to continue to guide Apple successfully (but it is by no means certain).
It's not different than my kids following my beliefs if I died today.
