User talk:Martin Hogbin/MHP - Complete and consistent solutions

This is a page where I try to explain to anyone interested which solutions to the Monty Hall Problem I consider complete and consistent. I start by giving a version of the problem which clearly asks a conditional question. The most well known version of the problem, by vos Savant and Whitaker, is somewhat ambiguous in this respect.

Game rules
We adopt the standard game rules, that the host must always open an unchosen door to reveal a goat and must always offer the swap.

Assumptions
If we are Bayesians we must take the location of the car, player's original door choice, and the host's legal door choice to be uniform at random because we have been given no information that might enable us to make any other assumptions.

Frequentists normally specify these distributions to be uniform in the standard version of the problem.

C - There is a symmetry with respect to door numbers
The door numbers can be arbitrarily rearranged at the start of the game without changing any probabilities.

Simple solutions
It can be argued that the simple solutions are incomplete because they do not allow for the possible choice of door by the host when the player has originally chosen the car.

The simple solutions can be made complete by adding observations A and B or just observation C.

'Conditional' solutions
It can be argued that the 'conditional' solutions are incomplete because they do not account for the player's initial door choice. The conditional solutions can be made complete by adding either observation A or observation C.