 # Description The game Monopoly is played on board with 40 squares, ...

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Description The game Monopoly is played on board with 40 squares, set up as shown below GO A1 CC1 A2 T1 RR1 B1 CH1 B2 B3 JAII H2 C1 T2 UT1 H1 C2 CH3 C3 RR4 RR2 G3 D1 CC3 CC2 G2 D2 G1 D3 G2J F3 UT2 F1 F1 RR3 E3 E2 CH2 El FP Players begin the game on the GO square. On each turn, the player rolls two 6-sided dice. The sum of the dice determines the number of squares they advance (in clockwise direction) on that turn. Without any further rules. we would expect players to visit each square with equal probability However landing on G2J (go to jail), CC (community chest) and CH (chance) changes this distribution. When a player lands on G2J, they must immediately move tothe JAIL square. At the beginning of the game, the CC and CH cards are shuffled When player lands on CC or CH they take card from the top of the respective pile and, after following the instructions. returned to the bottom of the pile. There are 16 cards each pile but for this assignment are only concerned with cards that order movement Any card not concerned with movement will be ignored and the player will remain on the CC/CE square. The relevant card: are: Community Chest (2/16) Advance GO 2. Goto JAIL Chance(10/16) Advance GO 2. GotoJAIL 3. Goto( C1 4. Goto E3 5. GotoH2 6. GotoRi 7. Goto next RR (railroad) 8 Goto next RR 9 Goto next UT (utility) 10. Goback 3 squares In addition to G2J and CC/CH if player rolls doubles con three consecutive turns, they proceed directly to jail. A double" is : roll where the two dice are equal, such as rolling 2 fives or sixes. The goal of this assignment is to estimate the long- term prohabilities of landing on each square That is to compute for each square the probabilty that the player will end turn that square if they play the game for infinitely many turns. This assignment is based on Project Euler exercise Questions Use R to find answers to all of the following questions (that is. don't do any by hand by point-and-click) Save your code in an script. 1. Write function simulate _monopoly() that simulates turns by player in game of Monopoly using two d-sided dice The inputs yoyr function should ben and d. The output of your function should bez length n+ vector of positions, encoded as numbers from to 39 2. Writeafunction est imate monopoly< that uses your simulation toestimate the long term probabilitie of ending turn on each Monopoly square. What are the most likely squares to end turn on if you play Monopoly with 6-sided dice? What you play with 4-sided dice? Display graphically the long-term probabilities for 3. 4.5. and 6-sided dice. 3. Usse 000 simulations with n = 10. 000 turns each to estimate the standard error for the long-term probability of ending turn jail. 4. Use the non-parametric bootstrap with = 1,000 samples from simulation of 10,000 turns to estimate the standard error for the long-term probability of ending turn jail. (a) How does the bocestrap estimate compare to the simulation estimate? Which do you think is more accurate? Explain (b) Which faster to compute: the bootstrap estimate or the simulation estimate? Explain why there difference. 5. Display graphically the standard errors for the long-term probabilities for 3, 4.5. and 6-sided dice (use the same settings you used in question 2). Discuss why some probabilities have much larger standard errors than others 6. What happens to the standard errors for the long-term probability estimates as n increases? Why does this happen?

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s1 <- c("GO","A1","CC1","A2","T1","RR1","B1","CH1","B2","B3","JAIL","C1","UT1","C2","C3","RR2","D1","CC2","D2","D3")
s2 <- c("FP","E1","CH2","E2","E3","RR3","F1","F2","UT2","F3","G2J","G1","G2","CC3","G3","RR4","CH3","H1","T2","H2")

squares <- c(s1,s2)

CHcard <- function(x,i,deck) {
drawn <- deck[(i %% 16)+1]
if (drawn<=6) {c(0,10,11,24,39,5)[drawn]}
else if (drawn==7 | drawn==8) {(ceiling(0.1*x+0.5)*2-1)*5 %% 40}
else if (drawn==9) {if (x>12) {12} else {28}}
else if (drawn==10) {(x-3) %% 40}
else {x}
}

CCcard <- function(x,i,deck) {
drawn <- deck[(i %% 16)+1]
if (drawn<=2) {c(0,10)[drawn]}
else {x}
}

next_real <- function(x,cc.i,ch.i,cc.deck,ch.deck) {
if (x==7 | x==22 | x==36) {CHcard(x,ch.i,ch.deck)}
else if (x==2 | x==17 | x==33) {CCcard(x,cc.i,cc.deck)}
else if (x==30) {10}
else{x}
}

roll <- function(d) {
rr <- floor(runif(6,min=1,max=d+1))
check <-c(rr==rr,rr==rr,rr==rr)
if (sum(check)==3) {0}
}

simulate_monopoly <- function(n,d){
seq <- c(0)
CC.i <- 1
CH.i <- 1
CCdeck <- sample(1:16)
CHdeck <- sample(1:16)

for (i in 2:(n+1)){
if (roll(d)==0) {seq[i] = 10}
else {
next_tntve = (seq[i-1] + roll(d)) %% 40...

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