Sandbox

Tuesday’s child

The famous nursery rhyme proclaims: “Tuesday’s child is full of grace.” Well, factoring in Tuesday for a birth date as discussed in Chance News 64 and Andrew Gelman’s blog produced a flood of comments.

Without Tuesday muddying the waters, the well-known answer to

I have two children.

One is a boy.
What is the probability that I have two boys?

is 1/3, rather than 1/2 as many are prone to say. William Feller in his famous book (Introduction to Probability Theory and Its Applications, Volume I, Third Edition, page 117) says the value of 1/2 is the solution to a much simpler problem: “A boy is chosen at random and comes from a family with two children; what is the probability that the other child is a boy?” He explains why: The 1/3 “might refer to a card file of families,” while the 1/2 “might refer to a file of males. In the latter, each family with two boys will be represented twice, and this explains the difference between the two results.”

Many of the comments focused on the intuitively irrelevant aspect of Tuesday and yet, a careful laying out of the sample space indicates that the day of the week for the birth of a boy turns out to be relevant. Some of the comments tried to explain the cognitive dissonance by referring to similarities to the so-called Monty Hall Problem, in the sense that available information needs to be accounted for.

With Tuesday thrown into the mix, the answer to

I have two children.

One is a boy born on Tuesday.
What is the probability that I have two boys?

surprisingly, turns out to be 13/27, which is close to 1/2, the answer to the simpler problem.

Consider a different physical situation, where "boy" now represents a successful knee operation and "girl" now represents an unsuccessful knee operation--we have, after all, but two knees. Ignoring the "Tuesday" aspect, knowing there is a successful knee operation implies a 1/3 chance of two successful knee operations. But this seems especially the wrong-way round because knowing of an unsuccessful knee operation implies a 2/3 chance of a successful knee operation.

When "Tuesday" is added to knee replacement, the implication is closer to 1/2. In fact if we recorded time of day to the nearest minute of the day, rather than to the particular day of the week, we would be even much closer to 1/2. But that is bothersome too because this allows for manipulation of the data keeping/presentation merely by tacking on what might be deemed a "spurious" variable that can take on many values. Expanding on Feller’s explanation, what is the proper “card file” to use?