Week 10 Homework

Smith is in jail and has 1 dollar; he can get out on bail if he has 8 dollars.A guard agrees to make a series of bets with him. If Smith bets A dollars,he wins A dollars with probability .4 and loses A dollars with probability .6. Find the probability that he wins 8 dollars before losing all of his money if: (a) he bets 1 dollar each time (timid strategy). (b) he bets, each time, as much as possible but not more than necessary to bring his fortune up to 8 dollars (bold strategy). (c) Which strategy gives Smith the better chance of getting out of jail?

# Function to simulate one round of betting
simulate_bet <- function(wager) {
  outcome <- sample(c(-wager, wager), size = 1, prob = c(0.6, 0.4))
  return(outcome)
}

# Function for the timid strategy
timid_strategy <- function(starting_money, bail_amount, bet_amount) {
  money <- starting_money
  while (money < bail_amount && money > 0) {
    money <- money + simulate_bet(bet_amount)
  }
  return(money >= bail_amount)
}

# Function for the bold strategy
bold_strategy <- function(starting_money, bail_amount) {
  money <- starting_money
  bet_amount <- min(8 - money, money)
  while (money < bail_amount && money > 0) {
    money <- money + simulate_bet(bet_amount)
    bet_amount <- min(8 - money, money)
  }
  return(money >= bail_amount)
}

# Simulate timid strategy
timid_wins <- replicate(10000, timid_strategy(1, 8, 1))
timid_probability <- mean(timid_wins)

# Simulate bold strategy
bold_wins <- replicate(10000, bold_strategy(1, 8))
bold_probability <- mean(bold_wins)

cat("Probability of winning with timid strategy:", timid_probability, "\n")

## Probability of winning with timid strategy: 0.0197

cat("Probability of winning with bold strategy:", bold_probability, "\n")

## Probability of winning with bold strategy: 0.0644

if (timid_probability > bold_probability) {
  cat("Timid strategy gives Smith a better chance of getting out of jail.\n")
} else if (bold_probability > timid_probability) {
  cat("Bold strategy gives Smith a better chance of getting out of jail.\n")
} else {
  cat("Both strategies have the same chance of getting Smith out of jail.\n")
}

## Bold strategy gives Smith a better chance of getting out of jail.

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