Master Angela and Inspiration: Our Solution?

After exchanging tokens with Master Angela, I actually felt the will of investigating the issue of conditional probability further. The fellows from her extract, mentioned as (Selvin, 1975b), seem to have put a bit of effort in getting their results, but they seemed to completely disagree with our intuition.

We actually concluded, a bit ironically, since that opposes our first impressions and our printed opinion (Printed), that swapping is indeed a better strategy in this game.

It is, as Priest would put it, counter-intuitive.

When we wrote things from another perspective, all was revealed. See:

First Choice	Truth	Swap	Stick
Door 1	Door 1	Lost	Won
Door 1	Door 2	Won	Lost
Door 1	Door 3	Won	Lost
Door 2	Door 1	Won	Lost
Door 2	Door 2	Lost	Won
Door 2	Door 3	Won	Lost
Door 3	Door 1	Won	Lost
Door 3	Door 2	Won	Lost
Door 3	Door 3	Lost	Won

We have a total of 9 cases in this analysis. From this 9 cases, we get 18 possible situations because we eliminate one door and consider swapping and sticking. Considering the column that says Swap, we win 6 times. Considering the column that says Stick, we win 3 times. In this way, it is actually true that swapping is a better strategy than sticking each and every time.

The confusion that happens here is then that this is Optimization, but people who do Combinatorics would see things differently. If they go for their usual reasoning,  they are thinking of the person at that very moment, and this is pretty hard to explain, like the person will be without any knowledge of what is happening in the overall when making a decision, since Combinatorics is thinking of that moment only. People from Optimization will be thinking of the entire list of possible results as if the events have already occurred when they analyse things, so that they are seeing things from the perspective of the manager or strategist, if that makes sense. They want a strategy that is best for the game as a rule: Swapping or sticking. The person from Combinatorics wants to know their chances when swapping or sticking at that very moment, not a strategy they could adopt as a rule in terms of swapping or sticking.

Basically, this is not a problem for Combinatorics, but for Optimization or managerial sciences instead. The way we study things is different.

If all we have to do is making a choice at that very moment, all we know leads us to think that we have a 50 to 50 chance of winning if we swap or stick. That is right reasoning.

Priest would be wrong when suggesting that we should change Combinatorics because it would be wrong in the foundations. We also would be wrong when stating that we can simply apply its rules to this problem.

If we can study the whole set of possibilities, sit, and then come up with an answer, then we know that the best strategy is swapping.

Notice that Combinatorics works by cases, but, in this case, with the analysis in the way we drew it, we don’t really have cases. In the same line, and therefore in the same case, we have Swap and Stick, Won or Lost.

To get one case for each situation, we would have to organise things in a different way, so say:

First Choice	Truth	Win
Door 1	Door 1	Stick
Door 1	Door 2	Swap
Door 1	Door 3	Swap
Door 2	Door 1	Swap
Door 2	Door 2	Stick
Door 2	Door 3	Swap
Door 3	Door 1	Swap
Door 3	Door 2	Swap
Door 3	Door 3	Stick

We now have 9 cases. We now win by sticking 3/9 and we win by swapping 6/9, that is, 1/3 and 2/3. That is probably how the fellows got their result. Please read Equals to correct reasoning instead of believing this.

References

Wikipedia. (2016). Monty Hall Problem. https://en.wikipedia.org/wiki/Monty_Hall_problem

UAH.(2016). Conditional Probability. http://www.math.uah.edu/stat/prob/Conditional.html

Pinheiro, M. R. (2015). Words for Science. Indian Journal of Applied Research, 5(5). http://www.academia.edu/12181924/Words_for_Science

Pinheiro, M. R. (2016). Marcia Ricci Pinheiro. https://www.researchgate.net/profile/Marcia_Pinheiro4/publication/308256028_The_Monty_Hall_Show_and_Murphy's_Score/links/57df294408ae72d72eac21ab.pdf?origin=publication_detail

Pinheiro, M. R. (2016a). Monty Hall, Prof. Posamentier, and us.  https://drmarciapinheiro.wordpress.com/2016/09/29/monty-hall-prof-posamentier-and-us/

Pinheiro, M. R. (2013). The Monty Hall Problem and a few moments of shame for Modern Science and scientists: Newcastle, 2000, Australia. http://mathematicalcircle.blogspot.com.au/2013/09/the-monty-hall-problem-and-few-moments.html

Pinheiro, M. R. (2016b). The Monty Hall Show and Murphy’s Score. https://www.researchgate.net/profile/Marcia_Pinheiro4/publication/308256028_The_Monty_Hall_Show_and_Murphy's_Score/links/57df294408ae72d72eac21ab.pdf?origin=publication_detail