Bayesian Probability in Taxi Cab and Girl Wearing Trousers problem?
this is not strictly relevant to JASP but I have a question about two problems which aim to explain Bayesian probability to learners. I really hope someone can help me understand this better.
In Eirini Koutoumanou's UCL workshop, a problem is given whereby a student from a school is observed as wearing trousers. Eirini's workbook gives Pr(θ|data)=Pr(data|θ)Pr(θ) / Pr(data) as Bayes' Theorem. Because only 40% of students are girls, and half the girls wear trousers (50%), and all the boys wear trousers (60%) this gives us: Pr(observed student who wore trousers was a girl)=Pr(any student is girl) *Pr(any girl wears trousers) / Pr(any student wearing trousers) = 04.0.5/0.8=0.25.
In the taxi cab problem (Tversky & Kahnemann, 1982), 15% of all cabs in a city are blue, 85% are green. An accident happens where an eyewitness claims that the cab was blue. When tested for his/her eyesight, the witness correctly identifies a blue cab as blue 80% of the time and wrongly as green 20% of the time. They give a different formula for calculating the Bayesian probability: Pr(blue cab) / Pr (green cab) * Pr (eyewitness correct) / Pr (eyewitness wrong), which gives them a probability of 0.41.
However, if you switch the formulae between the problems, you get different results:
Taxi formula with Girl data: Pr(student is girl) / Pr (student is boy) * Pr (girl wears trousers) / (student wears trousers) = 0.41.
Girl formula with taxi data: Pr(cab is blue)Pr(eyewitness correct)/Pr(eyewitness correct and cab is blue)= 0.150.8/0.12=1.
Where have I gone wrong here?
The Tversky & Kahneman formulation expresses the odds, not the probability. If you write that equation out in full (including the = sign) you will see that the posterior odds is 0.7058824. To get to the associated posterior probability you need to go 0.7058824/1.7058824 = .41.
For instance, in the Taxi formula with the Girl data you have:
Pr(student is girl) / Pr (student is boy) * Pr (girl wears trousers) / (student wears trousers)
but the last term should be "boy wears trousers", not "student wears trousers"