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 A dollars with probability .6.
Find the probabiltiy that he wins 8 dollars before losting all of his money if a) he bets 1 dollar each time b) he bets, each time as much as possible but not more than necessary to bring his fortune up to 8 dollars c) which strategy gives him the better chance of getting out of jail
Below is scenario A - $1 bet
library(matrixcalc)
# Possible states: 0,1,2,3,4,5,6,7,8
#Absorbing states: 0 - broke, 8 - out of jail
P<-matrix(c(1,0,0,0,0,0,0,0,0,
.6,0,.4,0,0,0,0,0,0,
0,.6,0.4,0,0,0,0,0,
0,0,.6,0,.4,0,0,0,0,
0,0,0,.6,0,.4,0,0,0,
0,0,0,0,.6,0,.4,0,0,
0,0,0,0,0,0,.6,0,.4,0,
0,0,0,0,0,0,.6,0,.4,
0,0,0,0,0,0,0,0,1), byrow=TRUE,nrow=9,ncol=9)
P## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
## [1,] 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
## [2,] 0.6 0.0 0.4 0.0 0.0 0.0 0.0 0.0 0.0
## [3,] 0.0 0.6 0.4 0.0 0.0 0.0 0.0 0.0 0.0
## [4,] 0.0 0.6 0.0 0.4 0.0 0.0 0.0 0.0 0.0
## [5,] 0.0 0.0 0.6 0.0 0.4 0.0 0.0 0.0 0.0
## [6,] 0.0 0.0 0.0 0.6 0.0 0.4 0.0 0.0 0.0
## [7,] 0.0 0.0 0.0 0.0 0.0 0.6 0.0 0.4 0.0
## [8,] 0.0 0.0 0.0 0.0 0.0 0.0 0.6 0.0 0.4
## [9,] 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0initial<-matrix(c(0,1,0,0,0,0,0,0,0), byrow=TRUE,nrow=1, ncol=9)
initial## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
## [1,] 0 1 0 0 0 0 0 0 0P2<-initial%*% P
P2## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
## [1,] 0.6 0 0.4 0 0 0 0 0 0P3<-P2%*% P
P3## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
## [1,] 0.6 0.24 0.16 0 0 0 0 0 0P4<-P3%*%P
P4## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
## [1,] 0.744 0.096 0.16 0 0 0 0 0 0P5<-P4 %*% P
P5## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
## [1,] 0.8016 0.096 0.1024 0 0 0 0 0 0P6<-P5 %*% P
P6## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
## [1,] 0.8592 0.06144 0.07936 0 0 0 0 0 0Trans<-matrix.power(P,100)
Result<-initial %*% Trans
Result## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
## [1,] 1 5.949614e-15 7.230272e-15 0 0 0 0 0 0This tells me, over time the gambler will eventually be ruined.
However, what is the probability her will win before he loses?
Scenario A) Below is scenario A - $1 bet What is the probability he wins before losing?
# gamble(k, n, p)
# k: Gambler's initial state
# n: Gambler plays until either $n or Ruin
# p: Probability of winning $1 at each play
# Function returns 1 if gambler is eventually ruined
# returns 0 if gambler eventually wins $n
gamble <- function(k,n,p) {
stake <- k
while (stake > 0 & stake < n) {
bet <- sample(c(-1,1),1,prob=c(1-p,p))
stake <- stake + bet
}
if (stake == 0) return(1) else return(0)
}
k <- 1
n <- 8
p <- .4
trials <- 100000
simlist <- replicate(trials, gamble(k, n, p))
mean(simlist) ## [1] 0.97993#What is probability he will win
1-mean(simlist)## [1] 0.02007Below is scenario B - Max bet possible # Possible states: 0,1,2,4,8 #Absorbing states: 0 - broke, 8 - out of jail
Q<-matrix(c(1,0,0,0,0,
.6,0,.4,0,0,
0,.6,0.4,0,0,
0,0,.6,0,.4,
0,0,0,0,1), byrow=TRUE,nrow=5,ncol=5)
Q## [,1] [,2] [,3] [,4] [,5]
## [1,] 1.0 0.0 0.0 0 0.0
## [2,] 0.6 0.0 0.4 0 0.0
## [3,] 0.0 0.6 0.4 0 0.0
## [4,] 0.0 0.0 0.6 0 0.4
## [5,] 0.0 0.0 0.0 0 1.0initial2<-matrix(c(0,1,0,0,0), byrow=TRUE,nrow=1, ncol=5)
initial2## [,1] [,2] [,3] [,4] [,5]
## [1,] 0 1 0 0 0Q2<-initial2%*% Q
Q2## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.6 0 0.4 0 0Q3<-Q2%*% Q
Q3## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.6 0.24 0.16 0 0Q4<-Q3%*%Q
Q4## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.744 0.096 0.16 0 0Q5<-Q4 %*% Q
Q5## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.8016 0.096 0.1024 0 0Q6<-Q5 %*% Q
Q6## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.8592 0.06144 0.07936 0 0Trans2<-matrix.power(Q,100)
Result2<-initial2 %*% Trans2
Result2## [,1] [,2] [,3] [,4] [,5]
## [1,] 1 5.949614e-15 7.230272e-15 0 0This tells me, over time the gambler will eventually be ruined.
However, what is the probability her will win before he loses?
Scenario b) Below is scenario B - Max bet possible possible states: 0,1,2,4,8 Absorbing states: 0 - broke, 8 - out of jail
What is the probability of winning?
# gamble(k, n, p)
# k: Gambler's initial state
# n: Gambler plays until either $n or Ruin
# p: Probability of winning $1 at each play
# Function returns 1 if gambler is eventually ruined
# returns 0 if gambler eventually wins $n
gamble2 <- function(k2,n2,p2) {
stake2 <- k2
while (stake2 > 0 & stake2 < n2) {
bet2 <- sample(c(-stake2,stake2),1,prob=c(1-p2,p2))
stake2 <- stake2 + bet2
}
if (stake2 == 0) return(1) else return(0)
}
k2 <- 1
n2 <- 8
p2 <- .4
trials2 <- 100000
simlist2 <- replicate(trials2, gamble2(k2, n2, p2))
mean(simlist2)## [1] 0.93585#What is the probability he will win?
1-mean(simlist2)## [1] 0.06415#So Scenario A)
1-mean(simlist)## [1] 0.02007#Scenario b)
1-mean(simlist2)## [1] 0.06415#Both will end in ruin, however, scenario b provides a better strategy of getting out of jail