We generalise a result in Professor David Mackay’s book on inference. Bayes theorem also plays a crucial role in decision-making:
Let us consider a worked example which demonstrates how unintuitive results following from this 260-year-old theorem can be.
In the Game Show example, we can simplify Bayes’ Theorem by using a form that expands out all individual doors, and then cancelling off the unconditional probability of each door in numerator and denominator as they are each , where there are n doors. We consider the two distinct representative cases:
and where means that any m doors were removed at the usual intermediate stage after I chose a door. When we start with n doors, and one has been chosen, and that door happens to have the prize behind it, then the Game Show Host is free to remove m doors from the set of doors and so there are available ways for the host to do so.
The probability is therefore , since we equivocate between all the host’s options. For the other case, there are only doors for the host to choose m from, so the number of ways is , and the probability is therefore .
after some algebra I find that
Thus, our probability factor is given by:
This expression gives us back, from the original game, our game strategy factor of 2 times better if we shift door when and . The factor rises to better for shifting to another door for any and doors (all but one other than the one the player chose), and for the shift strategy a probability factor that tends to unity from above when and , i.e. only one door is removed.
An informal survey by Student-b has shown that quite often, intuition among a random sample of people asked, is lacking. Some will believe it is better to stick, and some say in the standard three-door game that it is slightly better to shift. The mathematics show that for positive n and m: it is always better to shift!