---
title: "Project #2"
author: "Milica Cudina"
date: "`r Sys.Date()`"
output: pdf_document
urlcolor: blue
---
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```{r echo=FALSE, warning=FALSE, include=FALSE}
knitr::opts_chunk$set(
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```
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\fbox{\parbox{\textwidth} {This project is inspired by the case study \\
  {\em ``Chuck A. Luck Wagers a Buck: Probabilistic Reasoning and the
    Gambler’s Ruin''} by Christopher M. Rump.}}

Consider the game of *Chuck-a-Luck*. Here is how it's played as told by Mr. Daniel Reisman of Niverville, New York: 

\textcolor{blue}{\bf``Three dice are rolled in a wire cage. You place a bet on any number
from 1 to 6. If any one of the three dice comes up with your number,
you win the amount of your bet. (You also get your original stake
back.) If more than one die comes up with your number, you win the
amount of your bet for each match. For example, if you had a \$1 bet
on number 5, and each of the three dice came up with 5, you would win
\$3.''}

---

## Problem #1 **(3 points)**
What is the probability that the gambler wins the game?

## Problem #2 **(5 points)**
Who has the long-run advantage in the game: the house or the gambler? By how much on the dollar?

## Problem #3 (8 points)
How would you change the amount of the
  winnings in every round of *Chuck-a-Luck* to make the game fair?
  Say that doubles pay $\$a$ per \$1 bet rather than $\$2$ and triples  pay $\$b$ per \$1 bet in stead of \$3. What is the condition on $a$ and $b$?

## Problem #4 (10 points)
Get three regular dice or use software to create
  simulated values of the outcomes of rounds of *Chuck-a-Luck*
  with the original reward system. Play $100$ independent rounds of
  the game, starting with a new dollar each time. What is the average
  of the gambler's winnings in these $100$ rounds?

## Problem #5 (2 points)
Take another look at your answers to the above
  problems. How do they relate to the
  Strong Law of Large Numbers? If you forgot about this theorem you studied in *Probability*, see
  [Wikipedia](https://en.wikipedia.org/wiki/Law_of_large_numbers).
  
## Problem #6 (10 points)
Now we are going to look at a particular
  gambler who arrives at the casino with \$2 in his pocket intending
  to play *Chuck-a-Luck*. At the beginning of every round of the
  game, he invests exactly \$1 (if he has the money). He continues to
  play round after round of the game until he either loses everything,
  or his wealth reaches at least \$4.

  Again, you can use real dice or software to play the game. Keep
  track of the gambler's wealth as the rounds of *Chuck-a-Luck*
  are repeated until the entire string of rounds is ended by either
  bankruptcy, or his wealth being at least \$4. Display the gambler's
  wealth trajectory graphically.
  
## Problem #7 (22 points)
Having completed the above single run,
  now repeat the same exercise $100$ times. More precisely, you can
  imagine letting $100$ gamblers enter the casino with \$2 each and
  play rounds of *Chuck-a-Luck* as above. This time you do not
  need to keep track of each gambler's wealth as the game-play
  progresses. For each gambler you should just record:
\begin{itemize}
\item the {\bf number of rounds of the game} it took them to either go
  bankrupt or emerge victorious with at least \$4;
\item whether the {\bf final result} was bankruptcy or not.
\end{itemize}

What is the average number of rounds the $100$ gamblers played? What
is the proportion of gamblers who finished with at least \$4 in their
pocket? Draw the histogram of the number of rounds played for the
players who went bankrupt only, the histogram of the number of rounds
played for the remaining players, and the histogram of the number of
rounds played for all players.