Seeing the unseen

More on what is not readily apparent.

“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?”

If you intuitively answered 50% (or less) you are not seeing the entire picture. If you answered 2/3, then congratulations, you got it.

Easy explanation: The gender of two siblings has four possibilities or outcomes: (Big brother, little brother), (Big sister, little sister), (Big brother, little sister), (Big sister, little brother). Three of those outcomes (the last three) has 1 or more sisters. Out of those 3 outcomes, 2 includes a brother, hence the answer is 2 out of 3.

This kind of problem is what Schumacher refers to as a convergent problem. Convergent problems asymptotically converge towards a solution. This means that ultimately, there are no unknowns and it can be solved. Science has a long tradition of dealing with convergent problems. Real life, however, are full of divergent problems that have no closed solutions. Witness what happens when those two worlds collide (rocket scientists on Wall Street). Problems with unknown unknowns are problems that must be lived through. It is problems such as trying to arrange for optimal happiness by going from poor to rich with the self-defeating outcome that this strategy ultimately makes you rich and thus no longer a user of the happiness-strategy. It is problems such as trying to run a country, a company or even your life.

People can be given a strategy and they can implement such a strategy in their life. However, a strategy for a divergent problem is not a solution. It is just a suggestion of tactics.

Jacob comments again:

Birth order has nothing to do with the original question. It was merely a way to count. Consider this then.

A man has two siblings. We know they can be either males (M) or females(F). There are 4 ways of randomly picking two siblings: MM, MF, FM, FF. Now, we know that at at least one of his siblings is female. Therefore both siblings can’t be males. This means that the MM outcome is excluded. This leaves MF, FM, FF all three of which as at least one female. The question is how many of those contain a male. Two of them! Therefore with the information given (two siblings, at least one female), the probability is 2/3. (MF+FM)/(MF+FM+FF).

Maybe it helps to consider the different question of what the probability is that the other sibling is ALSO female given that at least one of them is female: FF/(MF+FM+FF) or 1/3.

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Originally posted 2008-11-19 07:12:31.