DATA 605 - Assignment 10

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

1. he bets 1 dollar each time (timid strategy). Answer = 0.0203

states <- c("0", "1", "2", "3", "4", "5", "6", "7", "8")
trans_mat <- matrix(c(1, 0, 0, 0, 0, 0, 0, 0, 0, 
                              0.6, 0, 0.4, 0, 0, 0, 0, 0, 0,
                              0, 0.6, 0, 0.4, 0, 0, 0, 0, 0,
                              0, 0, 0.6, 0, 0.4, 0, 0, 0, 0,
                              0, 0, 0, 0.6, 0, 0.4, 0, 0, 0,
                              0, 0, 0, 0, 0.6, 0, 0.4, 0, 0,
                              0, 0, 0, 0, 0, 0.6, 0, 0.4, 0,
                              0, 0, 0, 0, 0, 0, 0.6, 0, 0.4,
                              0, 0, 0, 0, 0, 0, 0, 0, 1), nrow = 9, byrow = TRUE, dimnames = list(states, states))

intial_state <- c(0, 1, 0, 0, 0, 0, 0, 0, 0)

n <- 100000

for (i in 1:n){
  intial_state <- intial_state %*% trans_mat
}
intial_state

##              0 1 2 3 4 5 6 7          8
## [1,] 0.9796987 0 0 0 0 0 0 0 0.02030135

2. he bets, each time, as much as possible but not more than necessary to bring his fortune up to 8 dollars (bold strategy). Answer = 0.064

trans_mat <- matrix(c(1, 0, 0, 0, 0, 0, 0, 0, 0, 
                              0.6, 0, 0.4, 0, 0, 0, 0, 0, 0,
                              0.6, 0, 0, 0, 0.4, 0, 0, 0, 0,
                              0.6, 0, 0, 0, 0, 0, 0.4, 0, 0,
                              0.6, 0, 0, 0, 0, 0, 0, 0, 0.4,
                              0.6, 0, 0, 0, 0, 0, 0, 0, 0.4,
                              0.6, 0, 0, 0, 0, 0, 0, 0, 0.4,
                              0.6, 0, 0, 0, 0, 0, 0, 0, 0.4,
                              0, 0, 0, 0, 0, 0, 0, 0, 1), nrow = 9, byrow = TRUE, dimnames = list(states, states))

intial_state <- c(0, 1, 0, 0, 0, 0, 0, 0, 0)

for (i in 1:n){
  intial_state <- intial_state %*% trans_mat
}
intial_state

##          0 1 2 3 4 5 6 7     8
## [1,] 0.936 0 0 0 0 0 0 0 0.064

3. Which strategy gives Smith the better chance of getting out of jail? Answer = Bold Stratergy Provides Better Chance For Freedom.