In this assignment we’ll practice writing random variables using the game of Roulette. We’ll use them in the next assignment to analyze and understand the game.
A European roulette wheel has numbers 1-36 in red and black, and number 0 in green for a house win. The dealer spins the board, and the the ball lands randomly on one of these 37 numbers. Here’s a brief video introducing the rules of roulette.
In the code that follows, think of x as a random variable uniformly distributed on the integers 0 through 36, representing the possible values for a game of roulette. Here’s one way to produce a vector containing a sample of size n from x.
n = 100
roulette_values = 0:36
x = sample(roulette_values, size = n, replace = TRUE)(4 points) general code quality across all questions.
(3 points)
The elements of our sample space are the roulette numbers: {0, 1, …, 36}. In statistics, a random variable is a function that maps an element of the sample space to a real number. Let h(x) be a random variable representing the amount you win or lose for a bet on the high numbers defined as follows:
\[ h(x) = \begin{cases} -1 &\mbox{if } x \leq 18 \\ 1 & \mbox{if } 18 < x \end{cases} \]
Define a vectorized function high that accepts a sample vector x and implements h(x).
high = function(x)
{
win = (x > 18)
ifelse(win,1,-1)
}Hint: Use even(x) below as a template.
even = function(x)
{
win = (x %% 2 == 0) & (x != 0)
ifelse(win, 1, -1)
}(2 points)
Define a vectorized function low that accepts a sample vector x and returns the amount you win or lose if you bet one unit money on the low numbers, from 1 to 18. Hint: you lose if X = 0.
low = function(x)
{
win = (x <= 18) & (x >= 1)
ifelse(win,1,-1)
}(3 points)
Define a vectorized function straightup that accepts x, a sample vector, and d, a digit between 0 and 36, and returns the amount you win or lose in roulette if you bet one unit money on the single value d. Hint: The payout for a straight up bet is 35 to 1.
straightup = function(x,d = 1)
{
win = (x == d)
ifelse(win,35,-1)
}(5 points)
Pick another possible roulette bet that does not have a 1 to 1 payout, and implement an appropriately named vectorized function that accepts x, and returns the amount you win or lose in roulette if the ball lands in x. Write it in mathematical notation below as f(x).
Let f(x) be a random variable representing the amount you win or lose for a bet on the first dozen numbers {1,..,12} defined as follows:
\[ f(x) = \begin{cases} -1 &\mbox{if } x > 12 \\ 2 & \mbox{if } 1 \leq x \leq 12 \end{cases} \]
firstdozen = function(x)
{
win = (x >= 1) & (x <= 12)
ifelse(win,2,-1)
}(3 points)
Calculate the expected value of your winnings after a single play for each of the 4 betting strategies described above. It should be negative, because the house always wins.
single_play = data.frame(x)
bet = 50
digit = 2
# high bet
single_play$high = high(x)
high_win = nrow(single_play[single_play$high == 1,])
# low bet
single_play$low = low(x)
low_win = nrow(single_play[single_play$low == 1,])
# straight up bet
single_play$straightup = straightup(x,digit)
straightup_win = nrow(single_play[single_play$straightup == 35,])
# first dozen bet
single_play$firstdozen = firstdozen(x)
firstdozen_win = nrow(single_play[single_play$firstdozen == 2,])
single_play$Total_After_Each_Roll = rowSums(single_play[2:5])
total = data.frame(colSums(single_play,100))
# print summary
printSinglePlay = function(bet,payout,total,type,eachWin)
{
cat("Bet",bet,"dollar(s) on",type, "with payout",payout,fill=TRUE)
payoff = total[type,"colSums.single_play..100."]
cat("Ball hits on the number(s):",eachWin,"times.","\nThen you",ifelse(payoff >= 0,"win","lose"),abs(payoff*bet),"dollar(s).\n\n",fill=FALSE)
}
printSinglePlay(bet,"1:1",total,"high",high_win)## Bet 50 dollar(s) on high with payout 1:1
## Ball hits on the number(s): 50 times.
## Then you win 0 dollar(s).printSinglePlay(bet,"1:1",total,"low",low_win)## Bet 50 dollar(s) on low with payout 1:1
## Ball hits on the number(s): 47 times.
## Then you lose 300 dollar(s).printSinglePlay(bet,"35:1",total,"straightup",straightup_win)## Bet 50 dollar(s) on straightup with payout 35:1
## Ball hits on the number(s): 3 times.
## Then you win 400 dollar(s).printSinglePlay(bet,"2:1",total,"firstdozen",firstdozen_win)## Bet 50 dollar(s) on firstdozen with payout 2:1
## Ball hits on the number(s): 35 times.
## Then you win 250 dollar(s).payoff_total = total["Total_After_Each_Roll","colSums.single_play..100."]
cat("If you bet on all four,","you",ifelse(payoff_total >= 0,"win","lose"),abs(payoff_total*bet),"dollars.\n\n",fill=FALSE)## If you bet on all four, you win 350 dollars.# data frames
single_play## x high low straightup firstdozen Total_After_Each_Roll
## 1 34 1 -1 -1 -1 -2
## 2 8 -1 1 -1 2 1
## 3 24 1 -1 -1 -1 -2
## 4 35 1 -1 -1 -1 -2
## 5 26 1 -1 -1 -1 -2
## 6 34 1 -1 -1 -1 -2
## 7 30 1 -1 -1 -1 -2
## 8 16 -1 1 -1 -1 -2
## 9 22 1 -1 -1 -1 -2
## 10 28 1 -1 -1 -1 -2
## 11 8 -1 1 -1 2 1
## 12 11 -1 1 -1 2 1
## 13 7 -1 1 -1 2 1
## 14 25 1 -1 -1 -1 -2
## 15 24 1 -1 -1 -1 -2
## 16 31 1 -1 -1 -1 -2
## 17 35 1 -1 -1 -1 -2
## 18 17 -1 1 -1 -1 -2
## 19 1 -1 1 -1 2 1
## 20 20 1 -1 -1 -1 -2
## 21 6 -1 1 -1 2 1
## 22 14 -1 1 -1 -1 -2
## 23 22 1 -1 -1 -1 -2
## 24 30 1 -1 -1 -1 -2
## 25 4 -1 1 -1 2 1
## 26 34 1 -1 -1 -1 -2
## 27 27 1 -1 -1 -1 -2
## 28 0 -1 -1 -1 -1 -4
## 29 3 -1 1 -1 2 1
## 30 2 -1 1 35 2 37
## 31 10 -1 1 -1 2 1
## 32 36 1 -1 -1 -1 -2
## 33 23 1 -1 -1 -1 -2
## 34 30 1 -1 -1 -1 -2
## 35 28 1 -1 -1 -1 -2
## 36 7 -1 1 -1 2 1
## 37 19 1 -1 -1 -1 -2
## 38 24 1 -1 -1 -1 -2
## 39 6 -1 1 -1 2 1
## 40 30 1 -1 -1 -1 -2
## 41 13 -1 1 -1 -1 -2
## 42 21 1 -1 -1 -1 -2
## 43 2 -1 1 35 2 37
## 44 22 1 -1 -1 -1 -2
## 45 19 1 -1 -1 -1 -2
## 46 2 -1 1 35 2 37
## 47 21 1 -1 -1 -1 -2
## 48 10 -1 1 -1 2 1
## 49 11 -1 1 -1 2 1
## 50 1 -1 1 -1 2 1
## 51 11 -1 1 -1 2 1
## 52 0 -1 -1 -1 -1 -4
## 53 8 -1 1 -1 2 1
## 54 15 -1 1 -1 -1 -2
## 55 18 -1 1 -1 -1 -2
## 56 9 -1 1 -1 2 1
## 57 30 1 -1 -1 -1 -2
## 58 15 -1 1 -1 -1 -2
## 59 8 -1 1 -1 2 1
## 60 1 -1 1 -1 2 1
## 61 31 1 -1 -1 -1 -2
## 62 5 -1 1 -1 2 1
## 63 31 1 -1 -1 -1 -2
## 64 14 -1 1 -1 -1 -2
## 65 33 1 -1 -1 -1 -2
## 66 27 1 -1 -1 -1 -2
## 67 16 -1 1 -1 -1 -2
## 68 0 -1 -1 -1 -1 -4
## 69 5 -1 1 -1 2 1
## 70 33 1 -1 -1 -1 -2
## 71 8 -1 1 -1 2 1
## 72 6 -1 1 -1 2 1
## 73 33 1 -1 -1 -1 -2
## 74 13 -1 1 -1 -1 -2
## 75 25 1 -1 -1 -1 -2
## 76 26 1 -1 -1 -1 -2
## 77 36 1 -1 -1 -1 -2
## 78 24 1 -1 -1 -1 -2
## 79 20 1 -1 -1 -1 -2
## 80 19 1 -1 -1 -1 -2
## 81 9 -1 1 -1 2 1
## 82 6 -1 1 -1 2 1
## 83 1 -1 1 -1 2 1
## 84 12 -1 1 -1 2 1
## 85 25 1 -1 -1 -1 -2
## 86 31 1 -1 -1 -1 -2
## 87 10 -1 1 -1 2 1
## 88 31 1 -1 -1 -1 -2
## 89 3 -1 1 -1 2 1
## 90 27 1 -1 -1 -1 -2
## 91 17 -1 1 -1 -1 -2
## 92 4 -1 1 -1 2 1
## 93 26 1 -1 -1 -1 -2
## 94 6 -1 1 -1 2 1
## 95 20 1 -1 -1 -1 -2
## 96 9 -1 1 -1 2 1
## 97 33 1 -1 -1 -1 -2
## 98 16 -1 1 -1 -1 -2
## 99 26 1 -1 -1 -1 -2
## 100 36 1 -1 -1 -1 -2total## colSums.single_play..100.
## x 1781
## high 0
## low -6
## straightup 8
## firstdozen 5
## Total_After_Each_Roll 7