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COUNTERINTUITIVE PROBLEM, INTUITIVE REPRESENTATION
Blog posts about counterintuitive probability problems generate lots of opinions with a high probability.
Andrew Gelman and readers have been having a lot of fun with the following probability problem:
I have two children. One is a boy born on a Tuesday. What is the probability I have two boys? The first thing you think is “What has Tuesday got to do with it?” Well, it has everything to do with it.
DSN agrees with Andrew that one virtue of the “population-distribution” method is that it forces one to be explicit about various aspects of the problem, and in so doing, causes much confusion to disappear.
As a public service this week, Decision Science News presents the population-distribution representation of the problem (what it thinks of as the Gigerenzerian / Hoffragian / Peter Sedlmeier-ian representation of the problem) in a visual form.
To follow the logic, see Andrew’s post on how he solved the problem. Voila:
Red means “outside the reference class”. Yellow means “in the reference class but not boy-boy”. Green means “inside the reference class and boy-boy”.
Boy-boy in the reference class occurs with probability Green / (Green + Yellow) or 13 /27
To see why DSN calls these Gigerenzerian / Hoffragian / Sedlmeierian representations, see:
Sedlmeier, P. (1997). BasicBayes: A tutor system for simple Bayesian inference.
Behavior Research Methods, Instruments & Computers, 29(3), 328-336.
Gigerenzer, G., & Hoffrage, U. (1995). How to improve Bayesian reasoning without instruction: Frequency formats. Psychological Review, 102,, 684–704.
(Sorry for not using R, excel is just darn fast for some things)