Monty Hall Variations

Introduction:

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.