The Curious Monty Hall Problem

This one was hidden in my blogs folder as an invisible document for the last four years, discovered it today while reorganising the desktop ! Penned the article after reading the ‘The curious incident of a dog in the night-time’. Found the book an interesting read, partly because I could relate to the mannerism of the protagonist !

In the novel ‘The curious incident of a dog in the night-time’ Christopher Boone tells only one joke. And the joke says that Mathematicians are the ‘best’. Better than economists, and logicians.

Discounting the fact that it was a joke, please bear with the hair splitting analysis. Taken out of context ‘best’ can mean anything, it may mean mathematicians are best at solving partial differential equations which is true (tautological that is). But it may also mean mathematicians are best at understanding purposeful human action, which is not true because economists are best at that.

Now because I like the novel so much and the way the protagonists explains things, I would also explain things in his style of narration.

Beginning of Christopher Boone style narration

Some mathematicians think that they can apply Probability to unique events to make decisions. For example, in the novel ‘The curious incidents of the dog in the nighttime’ Christopher Boone tells an interesting story about The Monty Hall Problem (http://en.wikipedia.org/wiki/Monty_Hall_problem). The problem is about a game of luck which he tells to show that numbers are not straight and how intuition can be wrong, and this is called an illustration.

He uses probability to justify Marilyn vos Savant’s claim that “you should always change and pick the final door because the chances are 2 in 3 that there will be a car behind that door”. But this is so stupid because it is silly to say that the chances are 2 in 3, since picking up the door is a one off event.

End of Christopher Boone style narration (rather difficult I say!)

If you try to pin down the meaning of the statement ‘chances are 2 in 3’ you really won’t achieve anything but a feeling of futility because in the context of a specific event it does not have any meaning. It is of course true that numbers are not straight and I think to an extent it is because mathematicians want it to be that way.

The statement that “the chances are 2 in 3 that there will be a car behind that door” implies that the specific event of choosing the door is part of a class of events and that 2/3^rd probability implies a frequency of occurrence of a particular event#. But the truth is a person playing the game is not going to play it more than once. So in such a case, does a probability of 2/3^rd play any significance in the decision making process of the person? None whatsoever!

The non-applicability of concept of class probability to case probability situations can be generalized to all situations where a person is gambling. Statistics and probability calculations may comfort a gambler punting on a victory for Australia meeting Zimbabwe but he is still a gambler and the past performance of the two countries still has no bearing on their next match.

#To illustrate the concept of ‘class of events’ and ‘class probability’, visualize a small island where the total population is 1000. For sake simplicity, assume that the population has remained at this level for the last 100 years. It has been observed that for the last 100 years, a total of 10 people die every year on the island. We then say that the number of deaths is a class of events and the class probability is 1%. L.I.C. can come and insure the population against an event of death with a fair degree of certainty that every year it will have to pay to families of 10 out of 1000 people.

This is different from case probability, where there is no class of events and the uncertainty is on account of a lack of knowledge regarding all the factors determining an outcome. In the example of the Monty Hall problem, the uncertainty is arising on account of the players non-knowledge of what is behind the door – a goat or a car.