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library(kableExtra)

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• A French roulette has:
  • 37 colored and numbered pockets on the wheel.
  • 0 is green.
  • In number ranges from 1 to 10 and 19 to 28:
    • Odd numbers are red.
    • Even numbers are black.
  • In ranges from 11 to 18 and 29 to 36:
    • Odd numbers are black.
    • Even numbers are red.

wheel = read.csv("https://nkha149.github.io/stat385-sp2020/files/data/roulette.csv")
wheel = as_tibble(wheel)
kable(wheel) %>%
  kable_styling(bootstrap_options = "striped", full_width = FALSE)

• We will write an R function named roulette() that simulate a roulette, that is it has:
  • Input: 2 arguments bet and amount
    • bet: argument that takes one of the following options:
      • low(1-18) or high (19-36): A bet that the number will be in the chosen range.
      • red or black: A bet that the number will be the chosen color.
      • even or odd: A bet that the number will be of the chosen type.
      • first or second or third: A bet that the number will be in the chosen dozen: first (1-12), second (13-24), or third (25-36).
      • any number from 0 to 36.
    • amount: amount in dollars that you want to bet on. The default value for amount is 1.
  • Output:
    • The amount of money you gain/lose after the bet.
      • -amount if you lose the bet.
      • The amount of money win is calculated following the table below.
      • Make sure to include the $ dollar sign.

Simulations and Graphing

roulette = function(bet, amount = 1) {
  pick = sample(x = 0:(nrow(wheel) - 1), size = 1, replace = TRUE)
  low = 1:18
  high = 19:36
  red = c(1, 3, 5, 7, 9, 12, 14, 16, 18, 19, 21, 23, 25, 27, 30, 32, 34, 36)
  odd = seq(from = 1, to = 36, by = 2)
  black = c(2, 4, 6, 8, 10, 11, 13, 15, 17, 20, 22, 24, 26, 28, 29, 31, 33, 35)
  even = seq(from = 2, to = 36, by = 2)
  first = 1:12
  second = 13:24
  third = 25:36
  if ((pick %in% low) & (bet == "low")) {
    amount
  } else if ((pick %in% high) & (bet == "high")) {
    amount
  } else if ((bet == "red") & (pick %in% red)) {
    amount
  } else if ((bet == "black") & (pick %in% black)) {
    amount
  } else if ((bet == "even") & (pick %in% even)) {
    amount
  } else if ((bet == "odd") & (pick %in% odd)) {
    amount
  } else if ((bet == "first") & (pick %in% first)) {
    amount * 2
  } else if ((bet == "second") & (pick %in% second)) {
    amount * 2
  } else if ((bet == "third") & (pick %in% third)) {
    amount * 2
  } else if (pick == bet) {
    amount * 36
  } else {
    amount * -1
  }
}

• We want to estimate the probability of winning if we keep betting on red. To do that, we use simulation studies, that is running the roulette() function many many times and record the number of times we win (not have a negative total amount at the end of the game). The number of simulations n is 5000.

set.seed(385)
results = replicate(roulette(bet = "red", amount = 1), n = 5000)
length(results[results == 1]) / 5000 # probability of winning if we keep betting on red

## [1] 0.4862

• Similarly, we want estimate the probability of winning if we keep betting on the first dozen. The number of simulations n is 5000.

set.seed(385)
results2= replicate(roulette(bet = "first"), n = 5000)
length(results2[results2 == 2]) / 5000 # probability of winning if we keep betting on the first dozen

## [1] 0.3114

• Now, we want to estimate the expected value of amount of money we will have by the end of the game if we bet on red with $1. We will do the simulations for 10000 times where n = 10,000.

set.seed(385)
results3 = replicate(roulette(bet = "red", amount = 1), n = 10000)
1 * length(results3[results3 == 1]) / 10000 - 1 * (1 - length(results3[results3 == 1]) / 10000) # expected value of amount of money we will have by the end of the game if we bet on odd with $5

## [1] -0.0248

Interpretation of Results

It helps to remember the meaning of expected value to interpret the results of this calculation. The expected value is basically, a measurement of the average. It indicates what will happen in the long run every time that we bet $1 on red.

While we might win several times in a row in the short term, in the long run, we will lose over 2 cents on average each time that we play. The presence of the 0 and 00 spaces are just enough to give the house a slight advantage. This advantage is so small that it can be difficult to detect, but in the end, the player always loses