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Marilyn is wrong resolves the dice paradox

[Originally authored 7/9/05, Updated: 9/9/05]

Back in May I posted about a probability paradox that had got me thinking. Well as is common, particularly when I have so many other things to do, an unrelated, casual conversation has lead me on a long, educational path. In this case, the path led me by an introduction to Marilyn vos Savant, and some very interesting connections of previously separate areas of my own knowledge.

Of particular interest to the probability paradox was a website called "Marilyn is wrong", which attempts to catch out errors published by Marilyn vos Savant. A few concerned cases of assumption or ambiguity strikingly similar to the dice paradox in my original post, but then I came across this one:

Marilyn Ignores the Obvious Regarding Probability of Boys

The content of which not only parallels the error of ambiguity and assumption, but also concerns the same fundamental problem as in my dice paradox. Despite the blatant bias held by the website, the page makes for critical reading for anyone still interested in the dice paradox.

Curiously enough, there is a response site called "Marilyn is right" which also addresses the probability of boys problem, and makes for an interesting alternate view, albeit, with much the same conclusions.

The Women With Two Children

[Update 9/9/05]
I've now had a chance to roll the new ideas around in my head and feel an explanation is in order. I'm making it now because it all seemed to come clear in the shower this morning, a convenient place to think mathematics indeed.

At first reading, Marilyn's solution to the Probability of Boys problem appears to be at odds with my conclusion to the dice paradox. And certainly, her answer of 1/3 (or 2/3 if the question asked for the probability of a girl) is equivalent to the 2/11 solution to the dice paradox. I argued 1/6 was the dice solution, which would be equivalent to the 1/2 two children solution. But the resolution within the original "Probability of Boys" page (and the followup "Women With Two Children" page) was not the actual solution, but the discussion of the merits of both solutions. Here's the critical difference between the two solutions, put as succinctly as I can imagine this morning:

In the two children problem, we are given an existing situation. Here is a family with two children, at least one of which is a boy. That immediately suggests that all the families without a boy have been disregarded. The family has been chosen as one with at least one boy. The correct probability that the other child is a boy is only 1/3.

In the dice paradox, the situation is new: two dice are rolled. That immediately suggests there has been no prior selection made on the outcome. Two dice are rolled and one of them happens to be a '4'. Not, "here are two dice that were rolled, at least one is a '4' ". Such a subtle difference but so very different situations to analyse. The difference is so subtle when put into words, that the reason for all the difference of opinion on the matter is clear.

The difference is clear to me though, and I don't think the questions are ambiguous. In the family situation a particular case has been consciously selected out and presented for analysis. In the dice situation, two independent dice are rolled and their outcome, whatever it happens to be, is presented for analysis. The difference is this: the extra information about the value of one of the dice is presented *after* the random event occured, and therefore cannot influence the probabilities of the outcomes. The family is *chosen to support* the extra piece of information and therefore adds prior knowledge we can use to update our probabilities.

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