Week 10

The problem:

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?

The solutions:

There are two possible approaches to solve the problem

The first approach

involves simulation. One can simulate multiple rounds of the game to estimate the probability of winning for both timid and bold strategies and then compare the results.

# Function to simulate one round of the game
one_round <- function(initial_money, target_money, win_prob) {
  money = initial_money
  while (money > 0 && money < target_money) {
    bet = min(money, target_money - money)
    outcome = sample(c(-bet, bet), size = 1, prob = c(1 - win_prob, win_prob))
    money = money + outcome
  }
  if (money >= target_money) {
    return(TRUE)  # Smith wins
  } else {
    return(FALSE) # Smith loses
  }
}

# Function to simulate multiple rounds of the game and calculate the winning probability
simulate_game <- function(initial_money, target_money, win_prob, num_simulations) {
  wins = 0
  for (i in 1:num_simulations) {
    if (one_round(initial_money, target_money, win_prob)) {
      wins = wins + 1
    }
  }
  return(wins / num_simulations)
}

# Parameters
initial_money = 1
target_money = 8
win_prob = 0.4
num_simulations = 10000

# Simulate the game for timid strategy
prob_timid = simulate_game(initial_money, target_money, win_prob, num_simulations)
cat("Probability of winning with timid strategy:", prob_timid, "\n")

## Probability of winning with timid strategy: 0.0626

# Simulate the game for bold strategy
prob_bold = simulate_game(initial_money, target_money, win_prob, num_simulations)
cat("Probability of winning with bold strategy:", prob_bold, "\n")

## Probability of winning with bold strategy: 0.0642

# Strategy comparison
if (prob_timid > prob_bold) {
  cat("Timid strategy has a better chance of getting out of jail.\n")
} else if (prob_timid < prob_bold) {
  cat("Bold strategy has a better chance of getting out of jail.\n")
} else {
  cat("Both strategies have equal chances of getting out of jail.\n")
}

## Bold strategy has a better chance of getting out of jail.

Second approach:

Let’s approach the problem differently by directly calculating the probabilities rather than simulating multiple rounds of the game.

 # Function to calculate the probabilities for timid strategy
calculate_prob_timid <- function() {
  P <- numeric(19)
  P[8] <- 1
  
  for (k in 7:2) {  # Adjusted loop index
    P[k] <- 0.4 * P[k + 1] + 0.6 * P[k - 1]
  }
  
  p1 <- sum(P[1:7])
  return(p1)
}

# Function to calculate the probabilities for bold strategy
calculate_prob_bold <- function(p1) {
  P <- numeric(9)
  P[8] <- 1
  
  for (k in 7:2) {
    P[k] <- (0.4 * P[k + 1] + 0.6 * P[k - 1]) / (1 - p1)
  }
  
  return(P[2])
}

# Calculate probability for timid strategy
p1_timid <- calculate_prob_timid()

# Calculate probability for bold strategy
p1_bold <- calculate_prob_bold(p1_timid)

# Compare strategies
if (p1_timid > p1_bold) {
  cat("Timid strategy has a better chance of getting out of jail.\n")
} else if (p1_timid < p1_bold) {
  cat("Bold strategy has a better chance of getting out of jail.\n")
} else {
  cat("Both strategies have equal chances of getting out of jail.\n")
}

## Bold strategy has a better chance of getting out of jail.

cat("Probability of winning with timid strategy:", p1_timid, "\n")

## Probability of winning with timid strategy: 0.663936

cat("Probability of winning with bold strategy:", p1_bold, "\n")

## Probability of winning with bold strategy: 2.843335

Conclusion:

Both approaches point at the bold strategy as the winning strategy.