Monty Hall, Bayes’ Law, and Sally Clark
Random Thoughts and the Monty Hall Setup
In this post I would like to discuss the issues that arise when we make choices with random probabilities, and then have to make follow-up decisions regarding the same problem when we have slightly more information.
Most people are aware of the famous Monty Hall challenge. The set up involves three doors; two of which hide a goat each, the last door, however, hides a spanking new car that will be yours, should you be lucky enough to choose that door. The game begins with you making your best (random) guess, say door number one. The show host then proceeds to let you know that of the remaining two doors, the car is not behind door number two. You are then faced with the question, "Would you like to stick with door number one, or would you like to switch to door number three?" (Of course if you were in it for the goat, I suppose you could just end the game there and say door number two)
This problem was subject to much debate when this problem gained notoriety in 1991 through Marilyn von Savant's column, and several very smart people defended the wrong answer. Most people assume, that the probability of getting the car once you have it narrowed down to 2 doors is 50-50. They are wrong!
It turns out that switching doors will double your chance of getting the car. (The probabilities of winning are 1/3 if you don't switch, and 2/3 if you do). What the majority (myself included) fail to realize when they assume a 50-50 shot, is that when Monty reveals a door that does not contain the car, he has effectively changed the rules of the game by acting intelligently within it. The fact that he always tells you a door behind which there is no car means that you have slightly more information than in a randomly distributed model and you can use that extra information to your advantage.
For a more robust explanation of the Monty Hall problem, the mathematics department at my alma mater, UCSD has a fun little interactive site where you can play the situation out for yourself and learn precisely the numbers behind the story. (http://www.math.ucsd.edu/~crypto/Monty/montybg.html)
Bayes' Law and Sally Clark Case
When I look for a final thought on the Monty Hall problem what I come away with is the importance of recognizing the rules of the game have shifted. In statistics, understanding the dynamic nature of probabilities is extremely important to assessing an event’s odds. This idea is all wrapped up and described by an idea called Bayes' law.
One well-known real world example of a misunderstanding of Bayes' law arose in the U.K. during the trial of Sally Clark. Clark was accused, convicted and subsequently exonerated of killing her two infant children, who had initially been recorded as having died of Sudden Infant Death Syndrome (SIDS). One of the statements believed to have strongly swayed the jury in this case was a statistic given by a Professor Roy Meadow; he stated that the chance of two children dying of SIDS in the same household would be 1 in 73 million.
The Prosecutor's Fallacy
Meadow had taken the overall probability of a particular child dying of SIDS, roughly 1 in 8,500, and merely squared it, to obtain the chance that two particular children would die of SIDS. As you can probably tell, this is not the most reasonable approach. The problem with this method is that it assumes that the events, (the two SIDS deaths) are completely unrelated, or in statistics language, independent and identically distributed. (i.i.d.). The reality is that very few things in this world really are truly i.i.d. Context is highly important, and in the case of SIDS deaths, it turns out that if you have one baby die of SIDS, you actually have a much greater chance of losing a second child that way. A paper on this particular case describes the true odds of the second child born in a family where the first child died of SIDS:
"It is intuitively clear that an infant in a family which has already suffered a SIDS will be at increased risk of SIDS, because many genetic and environmental factors will be the same. The published data allow us to estimate the actual level of dependency. According to the CESDI report...a baby is 10 times more likely to be a SIDS victim if a previous sibling was a SIDS victim than if not."1
Bayes' law hadn't been taken into account and the justice system suffered for it. Statisticians in the UK also pointed out that beyond finding the probability of two SIDS deaths, a comparison should have been made to the odds of a double murder. This analysis was done, and it turned out that SIDS was between 4.5 and 9 to 1 times more likely to be the answer relative to homicide.
It is extremely important to understand what you are trying to solve and the variables that impact those events...much like my previous conversion 'freakonomics' post, understanding context is essential to determining the true odds of events occurring, and being able to act accordingly.
Does anyone have experience with dealing with these kinds of mistakes professionally? Feel free to comment below.
1. 322 R. Hill © Blackwell Publishing Ltd. Paediatric and Perinatal Epidemiology 2004, 18, 320–326, (http://www.cse.salford.ac.uk/staff/RHill/ppe_5601.pdf)
A great talk given by Peter Donnelly on statistics, juries, and the abovementioned case: http://www.ted.com/talks/lang/eng/peter_donnelly_shows_how_stats_fool_juries.html
Posted by: Tristan Cordier