
Probability and Statistics Basic Probability and Statistics Math Forum 
 LinkBack  Thread Tools  Display Modes 
September 8th, 2019, 08:35 AM  #1 
Newbie Joined: Mar 2017 From: Usa Posts: 7 Thanks: 0  Bayesian inference problem
I thought this concept was simple so I haven't studied it in depth, but apparently I was wrong. This is from "Thinking Fast and Slow". A cab was involved in a hitandrun accident at night. Two cab companies, the Green and the Blue, operate in the city.  85% of the cabs in the city are Green (15% are Blue)  A witness identified the cab as Blue. The court tested the reliability of the witness under the circumstances that existed on the night of the accident and concluded that the witness correctly identified each one of the two colors 80% of the time and failed 20% of the time. What is P(cab involved in accident was Blue)? 
September 8th, 2019, 11:57 AM  #2 
Senior Member Joined: Jun 2019 From: USA Posts: 213 Thanks: 90 
I've never studied Bayesian inference, but a quick glance at the Wikipedia page tells me this might be consistent with what it was saying. Please do let me know if this is the answer he gave. P(cab was green, identified as green) = 0.85*0.8 = 0.68 P(cab was green, identified as blue) = 0.85*0.2 = 0.17 P(cab was blue, identified as blue) = 0.15*0.8 = 0.12 P(cab was blue, identified as green) = 0.15*0.2 = 0.03 P(total) = 0.68+0.17+0.12+0.03 = 1 (phew!) P(cab was blue, given constraint identified as blue) = 0.12/(0.12+0.17) = 0.414. Basically, combining the high reliability of the witness with the low probability of it being a blue cab in the first place. 
September 8th, 2019, 04:13 PM  #3 
Newbie Joined: Mar 2017 From: Usa Posts: 7 Thanks: 0 
Yeah that looks right. I can usually figure out stuff like this intuitively, but I was stuck on this. I like to be able to "feel" the answer rather than just plugging in the digits. I'm still having a hard time "feeling" .12/(.12+.17), but I'm working on it thanks

September 8th, 2019, 06:53 PM  #4 
Senior Member Joined: Jun 2019 From: USA Posts: 213 Thanks: 90 
When you have eliminated the impossible, whatever remains, however improbable, must be the truth. Either he saw a green cab and was wrong about the colour, or he saw a blue cab and was right. Out of the two normally unlikely occurrences, the latter is a little less likely. ...Now, what if, on top of all that, someone determined there was a 5% chance the witness was lying? 

Tags 
bayesian, inference, problem 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Statistical Inference problem  ghostactivist  Advanced Statistics  0  April 15th, 2016 12:06 PM 
Statistical Inference  EW Chiew  Math  1  December 21st, 2014 07:08 AM 
need help on statistical inference  finalight  Algebra  9  April 5th, 2012 09:46 PM 
Logic inference  NoobForever  Applied Math  3  January 4th, 2011 05:54 AM 
Bayesian or not Bayesian; this is my problem!  Marione  Advanced Statistics  0  April 24th, 2008 02:51 PM 