contact@onbelief.orgCognitive Illusion and Conditioned Probability
(or: Can Arithmetic Take Precedence Over Intuition?)
One difficulty that most of us have is that some problems with a probabilistic answer do not have an intuitively obvious solution. At times the solution can actually seem counterintuitive. One classic example is a variation of the Monty Hall Problem. This problem has been described as a cognitive illusion in which "a lack of appropriate information representation" causes difficulties in producing the correct solution.
I mention this problem now as an example of why we should be suspicious of our intuitions. After reading about the Monty Hall Problem below you might like to consider the extend to which our cognitive processes generate illusions. There are occasions where arithmetic, number and logic should take precedence over adjectives, feelings and intuitions. Indeed the more we describe the world in detail through systematic observation or research the more this condition holds.
A Complication of The Monty Hall Problem: The Second Observer
Assume you are a contestant on a TV game show. You are given the choice of opening one of three doors to win a car. Behind one door is a car. Behind each of the other doors are goats. Your aim is to pick the door that has the car behind it.
You pick door No. 1 but do not open at that point. The game show host, who knows what is behind all of the doors, opens door No. 3 to deliberately show you that one of the other 2 doors hides a goat. He then says to you, " You can either keep the your original choice of door No. 1 or you can pick door No. 2 instead ". You now have 2 doors in front of you. Is it to your advantage to switch your choice and open the other door or is there an equal chance that either of the 2 remaining doors will have the car behind it? ( adapted from Wikipedia)
To complicate matters now think about a friend of yours coming into the room at this point. Your friend has no previous knowledge of the situation and is simply told that behind one of the 2 remaining doors is a car and behind the other there is a goat. You are now both standing side by side looking at the same doors at the same time. You would be correct in thinking that your friend had an equal ( or 50%) chance of winning the car by picking either door. Why does that situation not apply to you ?
Answer to the Monty Hall Problem
You know that the door you picked initially had only a 1/3 probability of being correct. You can therefore deduce that the other 2 doors between them had a joint 2/3 probability. The host has now deliberately shown you that one of them does not have the car. However the 2 doors that were not your initial choice still jointly have 2/3 probability. You can now be logically certain that that 2/3 applies to the remaining door that you have not picked. After the additional information the host has given you, is is wiser to switch your answer because the probability calculation has changed. Your original choice still has the same probability of winning because nothing has happened to it. However but the odds have increased by a factor of 2 for the other door. Given the new information you should change your belief and therefore your action. By acting on your change of belief you have doubled your chance of winning the car from 1/3 to 2/3.A Simple Visual Explanation of the Monty Hall Problem
Your friend does not know that the door you picked only had a 1/3 probability initially. From your friend's perspective it is rational to view each door as now having an equal 1/2 (or 50%) probability of concealing the car. It is therefore rational for your friend to behave differently from you. On the other hand it is rational for you to expect to have a 1/6 better chance than your friend given your additional prior information. Despite the fact that you have doubled your chances of winning you are only 1/6 better off than you friend because she started off with a higher probability. [2/3-1/2= 1/6 alternatively 1/3-1/2 =-1/6]
If you still do not believe this solution go this site at the University of South Carolina or another site at Hofstra University and run the computer simulations.
A Variant on the Monty Hall Problem
Clearly if the game show host does not know what is behind the door when he opens it, he is undertaking a probabilistically different action in the world. The conditioning of the contestant's calculation must therefore change to accommodate his action.
More general thoughts on the Monty Hall Problem
Additional prior knowledge that conditions our understanding of the probability distribution in real life problems should forever influence our outlook. Measures that help us assess underlying probabilities are useful when making some decisions.
To see how this conditioning of probabilities can apply in a real life situation go to: Confusion of the inverse at Wikipedia to read about the significance of a diagnostic test from a doctor's perspective. When you read the article it is crucial to bear in mind that you are given assumed real prior information about a malignancy occurring. In an actual test situation the doctor might not have such prior information. The only rational conclusion that the doctor can draw in the situation where she has an imperfect test and does not know anything about the prior probability is that she should minimise the use of random testing. This strategy might at first seem illogical because superficially it appears that you cannot know about the prior distribution unless you condition your knowledge through testing. However testing in any meaningful sense requires a "gold standard". This idea is further developed on another page on this site.
Clearly the way in which we represent or make mental models of situations influences the way we act in the world. In how many situations do we fall for cognitive illusions and act on them ? It seems likely that if an explanatory idea is culturally well established it is likely that we will tend to ignore even the possibility of it being illusory.
Extending Cognitive Illusion into Every Day Life
To watch the video and also read commentaries on this talk (click here)
For further reading on the Monty Hall Problem see:
http://www.cs.dartmouth.edu/~afra/goodies/monty.pdf
On the Nature of Belief
www.onbelief.org
Scotland, 12th October 2007 and thereafter
Copyright 2007 onwards
