# Monty asymptotically to both 67 percent and

Monty Hall Variations

Introduction:

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The Monty Hall problem is a well-known mathematical problem
of probability, which originated from the gameshow “Let’s Make a Deal.” The
problem involves 2 goats and a car which are each hiding behind their
respective door. The host – Monty – is aware of which door hides what, while
the contestant does not know. Assuming

The original Monty Hall problem has been tested and proven
many times, though the maths behind it is very simple.

When picking a door without switching you are given a 1/3
chance of finding the car amongst the 2 goats. This chance is then amplified to
become 2/3 if you decide to switch, this is because when switching we instead
want our original pick to be a goat rather than the car, so that when the other
goat is revealed we can only pick the car. Using this tree diagram, we can
clearly see how this is true.

I decided to test the problem through a simulation found on
the site: http://www.mathwarehouse.com/monty-hall-simulation-online/

After running the simulation 500 times on both switching and
not, we can see that both results match what we thought.

I repeated the simulation, this time noting each individual
change so I could make a graph showing how the percentage of cars uncovered
might curve asymptotically to both 67 percent and 33 percent.

Variations:

Conditional probability

P(A|B) = x

The Monty Hall problem involves 1 car (c), 2 goats (g), 3
doors (d), 1 opened door (o), and 1 picked door (p). There are 5 variables. But
in extending and generalizing the Monty Hall problem, only 4 variables need to
be considered. That is because, of the 3 variables c, g, and d, each can be
derived from the two others. From the fact that c +g = d, it follows that c = d
-g and g = d – c. Only 2 of the variables c, g, and d therefore need to be
considered. In what follows, c (cars).

If the Monty Hall problem is generalized to any number of
cars (c), doors (d), and opened doors (o), and only 1 door is picked, the
chance of getting the car (C) by switching (s) doors (Cs) is

and the factor by which one improves one’s chances of getting
the car by switching is

The number 1 in these expressions represents the number of
picked doors (p), which is fixed at 1.