Monty Hall Non Linear Algebra Problem

The Monty Hall problem gives our player 3 doors to pick from. 1 of the door has a brand new car while the other 2 doors has a goat. The player already picked one door as their selection. The host who knew which door has a car behind it, openned a door with a goat and then offer the player to switch the door selection or not.

Since the host knew which door has a car and pick the door with a goat, the probability of getting a car for the player increases after the host eliminated the incorrect choice via goat door selection. So if the player make their choice after the host revealation then their probability of winning increases while before the host revealation, the probability of winning for player is only 33%.

So by making their selection choice 1 more time after host revealed the Goat door, the player winning chance increases by 17% from 33% to 50%. The action of randomly selecting between 2 remaining doors regardless of what the initial choice was would increase the probability of winning.

However, if the player switches their choice from inital one to the remaining door not openned or selected yet, the chance of winning actually increases by 33% to 66.67%. In the movie 21, Ben Cambel made the Switching choice, got the extra credit from professor Micky and got selected into the Black Jack team.

So if the player keep their choice, the probability of winning a car stays the same at 1/3 or 33%, randomly select the next choice among 2 remaining closed doors then the winning probability is 1/2 or 50% and switch their choice to the new door not selected or openned yet then the probability of winning is 2/3 or 66.67%.

Learning system like OpenAI dota superbot uses the Estimated Winning Probability throughout every game action state to guide their Exploration for Optimal Actions. By playing Dota games million or billion of time and store action sequences that resulting in higher winning probability, the Dota bot slowly updates its winning sequence neural network. Since computer can calculate probability much faster, more accurately and for every action it takes, the OpenAI Dota bot was able to beat the best human team in the world.