## R Project: Monty Hall Problem Simulation

The Monty Hall Problem problem is loosely based on the American television show *Let's Make a Deal*, originally hosted by Monty Hall, and became famous as a question that appeared in Marilyn vos Savant's "Ask Marilyn" column in Parade magazine in 1990:

*Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?*

Marilyn's response was that the contestant should switch to the other door, suggesting that contestants who switch have a $2/3$ chance of winning the car, while contestants who stick to their initial choice have only a $1/3$ chance. Many readers of vos Savant's column refused to believe switching increased one's chance of winning the car. Indeed, 10,000 of them -- including nearly 1,000 with PhDs -- wrote to the magazine, with most of them claiming Marilyn was wrong.

As evidence Marilyn vos Savant was correct, let us simulate this scenario using R...

First, write an R function `original.door()`

that simulates the situation where the contestant sticks with his or her original randomly-chosen door. The value returned should be a boolean value that is `TRUE`

when the contestant wins the randomly placed car and `FALSE`

when he or she loses.

Then, write an R function `switch.door()`

that simulates the situation where the contestant switches to the door that was both not chosen originally and not opened by Monty Hall (who -- when he has as choice of doors to open -- picks one at random). Again, the value returned should be `TRUE`

when the contestant wins the car, and `FALSE`

otherwise.

Finally, write an R function `cars.won(n,strategy)`

that takes a number of trials $n$, and a strategy (in the form of one of the function names: `original.door`

or `switch.door`

), and returns a simulated number of cars won in those $n$ trials, using the strategy supplied.

Use your `cars.won(n,strategy)`

function to test both strategies under 3000 trials. Which one works better? Do you believe Marilyn now?