@Mikko – The question states: “A man has TWO siblings…” Not three! If you want to include the man himself in the count (three different persons), you obviously can’t make him a female, which you just did. Also, while you initially talk about permutations, you then go on to list the combinations. In statistic, combinations and permutations are not the same thing. You need the permutations where FM is not the same as MF.

Please read all the post-comments above. It seems that everybody makes the same mistakes and they’ve been covered and explained multiple times already.

I think George nailed it first. Mike Hunt also got it among others who teased it out.

The problem changes after you say there is a sister. It’s a trick question in the sense that the number of total outcomes changes halfway through the problem.

The math would imply one or the other person is misguided because the real argument is are there three total outcomes or two. When are you supposed to start calculating probability, before Jacob tells you there is a sister or after?

Ironic how this was posted as seeing the unseen. Math can be blinding.

@Sclass – Maybe the title should have been “Accounting for all the information”. In general, probabilities should be updated with each new piece of information.

You start with the population statistics (presuming it’s known, if not, the convention is to start with a uniform distribution). However, once you’re told more, you’re calculating P(X|new information+old information). This is Bayesian updating and people tend to be really bad about it for rare uncertain events. This is not even THAT problem.

I think most people who trip up fail to notice the “AT LEAST ONE” part. This is a statement about which parts of the total distribution are still valid outcomes.

“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?”

I am not going to reveal their gender
= they are sitting in two non-transparet boxes in front of us, box No.1 and box No.2

I will tell you that he has at least one sister
= we know that in one of the boxes a woman is placed

What is the probability that he also has a brother
= what is the probability that there is a man sitting in one of the boxes

Possibilities:
1W2M
1M2W
1W2W

Answer: 2/3

Question is would all agree with the translation?

My assumption was that the question was not about existing children, but about children yet to be born (aka. “What gender will the next child be?”).

Are twins/triplets included in the probabilities? (Sorry, it’s too soon after lunch for me to calculate!)

Before you tell me I’m wrong in that 2/3 of these people are going to be male, there is no way to predict this and this is not what you asked. 2/3 of them has an equal chance of being male or female. It can go either way. So it’s 50/50 as to the chance of them being a boy or girl.

After you know the third person’s gender then you will be able to say what is as a percentage 2/3 male or female. But not before you know the gender of person 3. The unknown gender of person number 3 is not a math question, it’s a genetic outcome of a birth that the first 2 people have no influence on.

Older or younger siblings have no outcome as their gender has no influence of the outcome of the third persons gender.Each birth is a tossup as to it’s outcome.

If you think otherwise please explain how my brother or sister determined my gender. You can’t…

Sorry I didn’t explain that right. In the first scenario, if sibling A is a girl. sibling B is a boy. If sibling A is a boy. Sibling B is a girl. So if we know what Sibling A is, we know what sibling B is. What are the options of Sibling A. 50/50 boy girl. THEREFORE Sibling B is also 50/50

In the second scenario, If sibling A is a girl. Sibling B CAN BE A BOY OR A GIRL. If sibling A is a boy, Sibling B is a girl. If Sibling A is a girl we don’t instantly know the gender of Sibling B. This means we have to add a probability. Where the was only once choice, we know have two. So it’s (1+1) + 1 possibilities. Two of which include a boy

We have 2 different approaches here, interpreting the question:

1- Sintetic: we see “family” as a whole system and we see 2/3 probability;

2- Analitic: we see an isolated event inside the system “family” and we see 1/2 probability.

If Jacob wanted to induce the sintetic approach, he could have asked:
“We have a large set of families with 3 kids, and 1 is male in all of them. In this set, what’s the probability of the existence of families with 2 brothers and one sister?”