The Monty Hall problem is a famous case study in the application of Probability, specifically the concept of Conditional Probability. In simple terms Conditional Probability is the probability that"something" will occur given the additional information that "something else" has occurred. If "something else" is known to have occurred, it may or may not affect the probability of "something".
Examples of the probability of"something" not being affected by "something else" are (a) the outcome of the n-th coin toss of a "truly fair coin" given the results of n-1 previous tosses, and (b) the outcome of a pending Powerball draw given the results of all previous Powerball draws, assuming the balls or machinery have not been physically degraded or altered.
Examples of probability of "something" actually being affected by "something else" are (a) Blackjack - the probability of drawing a card to a non-Blackjack hand and busting given knowledge of all previous cards in the shoe that been exposed, and (b) the Monty Hall problem.
What is the Monty Hall Problem?
It was an American TV game show from the 1950s. There was a host called Monty Hall. The player was given a choice of 3 doors. Behind the doors, hidden from view, were 2 donkeys and a car, with each chosen randomly by the game producers. The players' goal was to choose the door with the car behind it. (Let's assume the car had more value than a donkey.)
The player initially chooses a door of his/her preference - 1, 2, or 3. The doors remained closed at this point, however. Then Monty, with the knowledge of which door has the car behind it, opens one of the two other doors. That door Monty chooses will have a donkey behind it. If the door the player initially chose actually has the car behind it, there is a 50% probability Monty will choose to open one of the two other doors. If the door the player chose does not have the car behind it, Monty must then open the one remaining door that also does not have the car behind it.
At this point the player is given a choice - stick to the door he initially chose or change his choice to the remaining the other door that remains unopened. The required analysis the player must undertake is to determine whether it is more advantageous to change his door or remain with the initially chosen door.
As it turns out, the player is better off changing his door choice. The reason for this is the additional information of the door Monty opened. There is a 2/3 probability of winning the car by changing the door than remaining with the initially chosen door. The math, involving Bayes Theorem, is as follows. To briefly explain the notation, "P(C3|P1,M2)" means the probability the car is behind Door 3, given that the player initially chooses Door 1 and Monty opens Door 2.