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).

# for maximum win, the markov chain must be:- state_1: 1, state_2: 2, state_3: 3, state_4: 4, state_5: 5, state_6: 6, state_7: 7, state_8: 8.
p_win <- 0.4
p_loss <- 1 - p_win
state_1 <- 1
state_8 <- 8

timid_strategy <- (1 - ((p_loss/p_win)^state_1)) / (1-((p_loss/p_win)^state_8))

print(paste0("The probability that he wins 8 dollars before losing all of his money if he bets 1 dollar each time is ", timid_strategy))
## [1] "The probability that he wins 8 dollars before losing all of his money if he bets 1 dollar each time is 0.0203013481363997"
  1. he bets, each time, as much as possible but not more than necessary to bring his fortune up to 8 dollars (bold strategy).
# for maximum win, the markov chain must be:- state_1: 1, state_2: 2, state_3: 4, state_4: 8
state_1 <- 1
state_2 <- 2

step_1 <- (1 - ((p_loss/p_win)^state_1)) / (1-((p_loss/p_win)^state_8))
step_2 <- (1 - ((p_loss/p_win)^state_1)) / (1-((p_loss/p_win)^state_8))
step_3 <- (1 - ((p_loss/p_win)^state_1)) / (1-((p_loss/p_win)^state_8))

bold_strategy <-  step_1 + step_2 + step_3

print(paste0("The probability that he wins 8 dollars before losing all of his money if he bets, each time, as much as possible but not more than necessary to bring his fortune up to 8 dollars is ", bold_strategy))
## [1] "The probability that he wins 8 dollars before losing all of his money if he bets, each time, as much as possible but not more than necessary to bring his fortune up to 8 dollars is 0.0609040444091991"
  1. Which strategy gives Smith the better chance of getting out of jail?
    Bold strategy gives Smith the better chance of getting out of jail because with bold statregy, Smith has 61% chance of getting out of jail compared to 20% chance of getting out of jail with timid strategy.