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).
  2. he bets, each time, as much as possible but not more than necessary to bring his fortune up to 8 dollars (bold strategy).
  3. Which strategy gives Smith the better chance of getting out of jail?
  1. matrix and graph for 1 dollar each time.
state_names <- c(0:8)
prob <- matrix(c(1,.6,0,0,0,0,0,0,0,
                 0,0,.6,0,0,0,0,0,0,
                 0,.4,0,.6,0,0,0,0,0,
                 0,0,.4,0,.6,0,0,0,0,
                 0,0,0,.4,0,.6,0,0,0,
                 0,0,0,0,.4,0,.6,0,0,
                 0,0,0,0,0,.4,0,.6,0,
                 0,0,0,0,0,0,.4,0,0,
                 0,0,0,0,0,0,0,.4,1), nrow=9, byrow=TRUE)

row.names(prob) <- state_names
colnames(prob) <- state_names

prob
##   0   1   2   3   4   5   6   7 8
## 0 1 0.6 0.0 0.0 0.0 0.0 0.0 0.0 0
## 1 0 0.0 0.6 0.0 0.0 0.0 0.0 0.0 0
## 2 0 0.4 0.0 0.6 0.0 0.0 0.0 0.0 0
## 3 0 0.0 0.4 0.0 0.6 0.0 0.0 0.0 0
## 4 0 0.0 0.0 0.4 0.0 0.6 0.0 0.0 0
## 5 0 0.0 0.0 0.0 0.4 0.0 0.6 0.0 0
## 6 0 0.0 0.0 0.0 0.0 0.4 0.0 0.6 0
## 7 0 0.0 0.0 0.0 0.0 0.0 0.4 0.0 0
## 8 0 0.0 0.0 0.0 0.0 0.0 0.0 0.4 1
plotmat(prob, pos = c(9),
        relsize = .9,
        lwd = 1, box.lwd = .5,
        cex.txt = 0.8,
        box.size = 0.03,
        box.type = "ellipse",
        box.prop = 0.8,
        box.col = "light yellow",
        box.cex = .8,
        dtext = 2,
        arr.lwd = .5,
        arr.length=.05,
        arr.width=.05,
        endhead = TRUE,
        self.cex = .8,
        main = "")

Using the strategy defined on page 9 here

we can see the probabilities starting at each stage:

p = 0.4
q = 0.6
r = q/p

for (i in seq(1, 7 , 1)){
  print ((1-r^i)/(1-r^8))
}
## [1] 0.02030135
## [1] 0.05075337
## [1] 0.0964314
## [1] 0.1649485
## [1] 0.267724
## [1] 0.4218874
## [1] 0.6531324
  1. bets as much as possible each time

New matrix starting at $3

state_names <- c(0:8)
prob <- matrix(c(1,0,0,.6,.6,0,0,0,0,
                 0,0,0,0,0,0,0,0,0,
                 0,0,0,0,0,0,0,0,0,
                 0,0,0,0,0,0,0,0,0,
                 0,0,0,0,0,0,.6,0,0,
                 0,0,0,0,0,0,0,0,0,
                 0,0,0,.4,0,0,0,0,0,
                 0,0,0,0,0,0,0,0,0,
                 0,0,0,0,.4,0,.4,0,1), nrow=9, byrow=TRUE)

row.names(prob) <- state_names
colnames(prob) <- state_names

prob
##   0 1 2   3   4 5   6 7 8
## 0 1 0 0 0.6 0.6 0 0.0 0 0
## 1 0 0 0 0.0 0.0 0 0.0 0 0
## 2 0 0 0 0.0 0.0 0 0.0 0 0
## 3 0 0 0 0.0 0.0 0 0.0 0 0
## 4 0 0 0 0.0 0.0 0 0.6 0 0
## 5 0 0 0 0.0 0.0 0 0.0 0 0
## 6 0 0 0 0.4 0.0 0 0.0 0 0
## 7 0 0 0 0.0 0.0 0 0.0 0 0
## 8 0 0 0 0.0 0.4 0 0.4 0 1
plotmat(prob,
        relsize = .9,
        lwd = 1, box.lwd = 1,
        cex.txt = 0.8,
        box.size = 0.03,
        box.type = "ellipse",
        box.prop = 0.8,
        box.col = "light yellow",
        box.cex = .8,
        dtext = 2,
        self.cex = .8,
        main = "")

not sure exactly how to calculate the probabilities hereโ€ฆ