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==Tuesday’s child==
==Tuesday’s child==
Author Michael Shermer pens a column for SciAm and has written several useful books on "skepticism" and science. 


[http://answers.google.com/answers/threadview?id=145411 The famous nursery rhyme] proclaims: “Tuesday’s child is full of grace.”  Well, factoring in Tuesday for a birth date as discussed in [http://www.causeweb.org/wiki/chance/index.php/Chance_News_64#A_probability_puzzle Chance News 64] and [http://www.stat.columbia.edu/~cook/movabletype/archives/2010/05/hype_about_cond.html Andrew Gelman’s blog] produced a flood of comments. 
In his recent book (''The Mind of the Market'', 2008) on how evolution has shaped human economic behavior, Shermer writes:


Without Tuesday muddying the waters, the well-known answer to
<blockquote>
<blockquote>I have two children.<br>
[More important], gamblers also tend to underestimate the number and length of winning streaks and lose out on the reward of placing larger bets during them.  Of course, even with an optimal betting strategy that plays to win every hand, and keeping loss aversion in check, if you play long enough you will lose because of the slight edge to the house built into the rules of the game. But casinos make even more money than the house percentage would predict because of our loss aversion.
One is a boy.<br>
What is the probability that I have two boys?
</blockquote>
</blockquote>
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
<blockquote>I have two children.<br>
One is a boy born on Tuesday. <br>
What is the probability that I have two boys?
</blockquote>
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. 
'''Discussion'''
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?

Revision as of 20:26, 20 July 2010

Tuesday’s child

Author Michael Shermer pens a column for SciAm and has written several useful books on "skepticism" and science.

In his recent book (The Mind of the Market, 2008) on how evolution has shaped human economic behavior, Shermer writes:

[More important], gamblers also tend to underestimate the number and length of winning streaks and lose out on the reward of placing larger bets during them. Of course, even with an optimal betting strategy that plays to win every hand, and keeping loss aversion in check, if you play long enough you will lose because of the slight edge to the house built into the rules of the game. But casinos make even more money than the house percentage would predict because of our loss aversion.

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