MMM (eliminated)
MMF
MFM
MFF
FMM
FMF
FFM
FFF (eliminated)

-Mikko

“A man has two siblings. I am not going to reveal their gender, but I will tell you that he has at least one sister. What is the probability that he also has a brother?”

As I see it, these are the permutations for the three siblings:

MMM
MMF
MFF
FMM
FMF
FFF

After we eliminate MMM and FFF (“man” is a man, and one of his siblings is female), we are left with these possibilities:

MMF
MFF
FMM
FMF

So there’s an equal probability for two females or two males.

Am I missing something?

-Mikko

I’m afraid I still don’t agree on that — the problem was as stated and it includes sufficient information to make the right conclusion, as well as the opportunity to make the wrong one (as I did initially, I admit…)

But more tongue-in-cheek though, I’d say that once we come to public policy decisions ALL problems are likely to be badly stated

“The whole point of the exercise is to point out that we shouldn’t be led to believe that. We’re easily misled by it and tend not to fully account for the ‘at least’, but this is what makes it conditional.”

That may be the point as you see it but it is not obvious from the wording of the original question.

“We are NOT dealing with the chances of a SINGLE sibling being of either gender, we are dealing with a population of two children, which always has a probability distribution of [snip]”

That is where I disagree. The way the problem is phrased, the “at least” is meaningless – the third sibling has to be either B or G, a binary situation. You are reading into it that the “at least” is supposed to be the clue that we are dealing with a population of more than three individuals. Not necessarily. If that was to be the case, then state it plainly (e.g., “…considering the normal male/female distribution in the human population…”). If that is a qualification/condition, then I agree, the odds are 2/3. But unless specifically conditioned as such we must confine ourselves to a population of three.

“I suspect the reasons Jacob wanted to develop this into healthcare later is that it’s a classic example of how badly we (and policy makers) can extrapolate partial information on a population level, possibly leading into marginal utility of healthcare expenditure when driven by political decision making? But it’s only a suspicion, and I still hope to persuade you to continue with that train of thought”

The problem is not partial information, it is a badly stated problem. By stating the original proposition as he did he limits the scope of the population to three.

But this exercise is useful since it points out something that any engineer or scientist knows (or should know) intutively, that is, most errors in solving a problem are the direct result of a misunderstanding of the problem. In this case, the misunderstanding was a direct result of a defective statement of the problem.

To quote your quote:

“This answer is intuitive if the question leads the reader to believe that there are two equally likely possibilities for the sex of the second child ”

The whole point of the exercise is to point out that we shouldn’t be led to believe that. We’re easily misled by it and tend not to fully account for the ‘at least’, but this is what makes it conditional.

We are NOT dealing with the chances of a SINGLE sibling being of either gender, we are dealing with a population of two children, which always has a probability distribution of 25% GG, 50% BG/GB and 25% BB. We have just enough data to rule out BB as impossible in this situation, so the conditional probablity is 2:1 in favour of BG over GG, therefore the sibling unaccounted for on the data given is twice as likely to be male than female.

The 1:2:1 distribution in the population derives directly from each individual independently having equal odds for either gender, and a known population size of 2.

I suspect the reasons Jacob wanted to develop this into healthcare later is that it’s a classic example of how badly we (and policy makers) can extrapolate partial information on a population level, possibly leading into marginal utility of healthcare expenditure when driven by political decision making? But it’s only a suspicion, and I still hope to persuade you to continue with that train of thought, Jacob

“The intuitive answer is 1/2. This answer is intuitive if the question leads the reader to believe that there are two equally likely possibilities for the sex of the second child (i.e., boy and girl, and that the probability of these outcomes is absolute, not conditional.”[citations deleted]

The question as you worded it does, indeed, assume an equal likelihood of the third child being B or G. There is no conditional probability involved in the statement of the question. Whether the third child is B or G is a simple binary proposition and is independent of the sex of the other two siblings.

I suggest reading the wiki link above.

Am I missing something!? Keep up the brain teasers!

Are you aware of any situations where this type of math can be applied to gain a real-life edge?

Interesting read here: http://en.wikipedia.org/wiki/Boy_or_Girl_paradox

(0.25MF+0.25FM)/(0.25MF+0.25FM+0.25FF+.25FF) = .5

you don’t know if she was the first or second female either.

(0.25MF+0.25FM)/(0.25MF+0.25FM+0.25FF) = 2/3

And if you know that the first-born is a female then

(0.25FM)/(0.25FM+0.25FF) = 1/2.

I’d really recommend either writing the computer program or flipping two coins multiple times.

If you flip two coins, you’ll get
Heads-heads = 25%
Different = 50%
Tails-tails = 25%

If you keep track of the coins
First heads, second heads = 25%
First heads, second tails = 25%
First tails, second heads = 25%
First tails, second tails = 25%

I don’t know if the confusion stems from being unable to see the difference between MF and FM or the difficulty of translating that into words.

So suppose you look at these 4 cases and is told that at least one is heads. This happens in 75% of the cases. Out of those 75%, how often does it happen that one shows tails. 50% out of 75% or 2/3.

That “at least one is a girl” is a valuable piece of information that changes the conditional probability. If you weren’t given this information, the answer would be 50%. If you were given the information that “one is a girl”, the answer would also be 50%. However, “one is a girl” is different from “at least one is a girl”. I think that’s what trips people.

This is not like the question where there are 3 doors, one of which has $1 million behind it, you pick one, and one of the other 2 is removed, do you switch doors?

These are independent events. If I could watch two roulette wheels, and the first one lands on Red, should I bet Black on the other one? Hint: NO

In your section where you commented again I believe the equation should actually be this:

(.5MF+.5FM)/(.5MF+.5FM+FF) = .5 not 2/3.

If you already know if the female was born first, then you can also exclude MF, and then it would be:

(FM) / (FM + FF) = .5

Two siblings. P(B=1 | G=1 or 2)?

That is I said
1) “A man has two siblings of unknown gender” (4 permutations, like a pair of dice have 36)
2) “at least one is a girl” (3 out of those permutations has at least 1 girl. Two of them has exactly 1 and one of them has 2.
3) Given that information, that is, there are three legal permutations, “how many has one boy in them” (answer 2).

Probability 2/3.

I think the question was pretty clear. You just have to be very precise in reading it. Every word is important. There’s a big difference between “at least one girl” and “one girl” for instance.

@spherical – If you want to do this computationally you need to
1) Generate a bunch of possible permutations of the two siblings assigning 50% probability that one is a girl and that one is a boy. You should get something like GG,BB,GB,BG,BG,GG,BG,GG,GB,GB,BG,BB …
2) Remove all those which has no girls in them keeping only those which have AT LEAST ONE GIRL (GG, GB, BG). This is your working list.
3) Count the fraction of the remaining list which has a boy in it. This should converge to 2/3.

I think in light of all the confusion, that’s MORE reason to do so

For me the take-away is ‘make sure you know what the question is before answering’ – and that it’s very easy to get it wrong! I leapt to the wrong conclusion initially, thinking we were assessing a single instance, but the framing refers to a set of instances – we’re analysing a population, not an individual.

while true:
for i = 0; i = 0.5)
print “B”
else
print “G”
println;

Nowhere in there do I increase the probability that a B will be generated if I’ve already generated a G.

*sigh* I guess this is either so subtle I can’t explain it or I really am missing something. Despite jabob’s explanation above, this really IS about combinations (order doesn’t matter). 66% probability of this being a boy/girl PAIR out of all possibilities (assuming order matters), 50% chance of the other sibling being a boy in this particular pair.

Thanks everyone. This was fun.

Stated one way: M has two siblings, one is F, so what is the probablilty of the other being F? Answer – 50/50. The reason is because you are asking us to determine the probability of a certain outcome in a binary system – M or F. This is the way you stated the problem.

Now stated another way: M has two siblings, one is F, so what is the probability that the one remaining sibling out of the three if F? Answer – 2/3. This is because you are asking us to determine the probability of a binary outcome (M or F) in a population of three. This is the way you THOUGHT you stated the problem.

The difference in phrasing is subtle but important.