Chapter 12.3 Prob 11

In American casinos, the roulette wheels have the integers between 1 and 36, together with 0 and 00. Half of the non-zero numbers are red, the other half are black, and 0 and 00 are green. A common bet in this game is to bet a dollar on red. If a red number comes up, the bettor gets her dollar back, and also gets another dollar. If a black or green number comes up, she loses her dollar.

  1. Suppose that someone starts with 40 dollars, and continues to bet on red until either her fortune reaches 50 or 0. Find the probability that her fortune reaches 50 dollars.
p <- 18/38
q <- 1 - p
N <- 40
M <- 10

probability_target <- (1 - (q/p)^N) / (1 - (q/p)^(N + M))
print(probability_target)
## [1] 0.3453043
  1. How much money would she have to start with, in order for her to have a 95% chance of winning 10 dollars before going broke?
target_probability <- 0.95

initial_fortune <- 1
max_initial_fortune <- 1000  

while (initial_fortune <= max_initial_fortune) {
  probability_target <- (1 - (q/p)^initial_fortune) / (1 - (q/p)^(initial_fortune + M))
  if (probability_target > target_probability) {
    break
  }
  initial_fortune <- initial_fortune + 1
}

if (initial_fortune <= max_initial_fortune) {
  print(initial_fortune)
} else {
  print("Initial fortune exceeds the maximum limit. Adjust the parameters.")
}
## [1] "Initial fortune exceeds the maximum limit. Adjust the parameters."
  1. A casino owner was once heard to remark that “If we took 0 and 00 off of the roulette wheel, we would still make lots of money, because people would continue to come in and play until they lost all of their money.” Do you think that such a casino would stay in business?

Even without green pockets on the roulette wheel, the casino would profit, as players tend to keep playing until they lose all their money. While removing green pockets might improve players’ odds slightly, casinos use strategies like setting limits and offering services to stay profitable. Thus, a casino without green pockets would likely remain profitable, though adjustments might be needed for financial stability.