User:Gene2024/sandbox

Visual Logic Tables
(The goal of this entry is to visually show the solution in a manner that is easy for people to understand)

To solve a problem, one must first understand it. Albert Einstein said, "If you can't explain it simply, you don't understand it well enough" and that "Genius is making complex ideas simple, not making simple ideas complex".

This is a relatively easy problem to solve if one takes the time to fully understand how the game works and what the rules are, and then incorporate that knowledge into the solution. Many make the problem more complicated than it is. There are probably more wrong solutions on the internet than there are correct solutions.

 The most common mistake 

The most common mistake seems to be the failure to properly understand and follow rule #2 in the “Standard assumptions” on the Wikipedia page: '“2. The host must always open a door to reveal a goat and never the car. ”' Keep in mind that the Host knows what is behind each door from the start of the game and therefore knows which door or doors can be opened without breaking the rules. Once the Host opens a door, it is not somehow magically changed to the other unopened door based on what's behind the door the Host opened. With the parameters that are defined in the problem (i.e.: the Guest selects Door #1 and the Host opens Door #3), the fact that the Host opens Door #3 means that the car is NOT behind Door #3 and the “Goat-Goat-Car” arrangement is not in play. The “Car-Goat-Goat” and “Goat-Car-Goat” are the only 2 arrangements that are in play.

 Marilyn vos Savant's solution  (WARNING: If you plan to try to solve the problem on your own, you may not want to read this or look at this table as it goes a long way towards defining the solution.)

Since Marilyn vos Savant's solution appears to be the de facto “gold standard” for this problem and most people are aware of and many followed her solution, it is a good solution to evaluate. As the saying goes "A picture is worth a thousand words" (attributed to Frederick R. Barnard), so Marilyn’s solution has been graphically entered in the “Marilyn vos Savant's Solution to the Wikipedia Monty Hall problem" table and has been formatted to reflect the parameters defined in the “Monty Hall problem”.  It is believed that it properly reflects the solution’s intensions, or at least what the results of the solution appears to indicate.

Because of the graphic way the table is presented and the additional information that was added, the problem with this solution becomes much easier to see: the solution from the Host opening Door #3 has been combined with a portion of the solution from Door #2, rather than just using the solutions from Door #3’s “Car-Goat-Goat” and “Goat-Car-Goat” configuration and voiding the “Goat-Goat-Car” solution since the Host opened Door #3 and the Door can’t be opened if the Car is behind Door #3.

There have been computer simulations written that support this solution, but at least some of them were likely influenced by this solution. It can be very difficult to write a program without being influenced by preconceived notions. A computer simulation can be written to support almost any (or all?) reasonable solutions, whether the solution is right or wrong.

 Visual Logic Table  (WARNING: If you plan to try to solve the problem on your own, you may not want to read this or look at this table as it fully defines the solution.)

To help eliminate any doubt that there may be inconsistencies in this table due to there being 1 Car and 2 Goats, this table shows all 6 possible Door Combinations (Guest chooses Door-X and Host opens Door-Y) and all 3 Games per Door Combination (Car-Goat-Goat, Goat-Car-Goat, and Goat-Goat-Car), along with which Games aren’t valid/playable (Games where the Car is behind the Door).

This table appears to comply to all the parameters and rules of the problem, and the results are consistent throughout all configurations and is predictable. It is therefore believed to be the correct solution. The table shows the odds of winning the car if the Guest stays with the door they originally chose, or switches to the unopened door. (It is not shown here as to not influence those who wish to solve the problem on their own.)