---
title: "Project #1: Part #2"
author: "Milica Cudina"
date: "`r Sys.Date()`"
output: pdf_document
urlcolor: blue
---
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<!-- If you are curious about what each option below does, -->
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```{r echo=FALSE, warning=FALSE, include=FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  fig.align="center",
  fig.pos="t",
  strip.white = TRUE
)
```
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---

Before embarking on this project, go to [Wikipedia: the Monty Hall problem](https://en.wikipedia.org/wiki/Monty_Hall_problem). This Wikipedia entry contains the description of the original Monty Hall problem as well as its analysis and variations. 

---

## Problem #1. **(10 points)**
Read and understand the description of the original Monty Hall game. Now, write down your own description in your own words. Resist the temptation to copy-paste from an external resource since that will earn you zero points. It might help to imagine that you have a five-year old niece or nephew you are describing the game to. 

## Problem #2. **(10 points)**
Write and execute code in `R` which will represent **one round** of the original Monty Hall game. More precisely, it should do the following:

- "choose" at random behind which door the prize will be,

- "choose" at random the door that the contestant picks,

- choose which door the host will open,

- determine whether it was optimal to **switch** or **stick** for this particular round and return this determination. 

## Problem #3. **(10 points)**
Write and execute code in `R` which repeats 100 rounds of the Monty Hall game. For which proportion of these rounds was it optimal to **switch**? *Hint:* If you answer to the previous problem was not in the form of a function, make it so now. 

## Problem #4 **(10 points)**
Write and execute code in `R` which repeats $n$ rounds of the Monty Hall game for $n=100, 110, 120, ..., 10000$. For each $n$, your code should record the proportion of individual rounds for which it was optimal to **switch**. In `R`, plot this recorded proportion as a function of $n$. 
Does your computational result agree with the analysis available at [Wikipedia: the Monty Hall problem](https://en.wikipedia.org/wiki/Monty_Hall_problem)? Why? It might also be useful to consult: [Wikipedia: The Law of Large Numbers](https://en.wikipedia.org/wiki/Law_of_large_numbers).