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?
Part a)
B = NR. Where N is the fundemental matrix and R is the canonical form. step 1: Create a matrix
a <- matrix(c(0,0.4,0,0,0,0,0,0.6,0,
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,
0,0,0,0,0,0.6,0,0,0.4,
0,0,0,0,0,0,0,1,0,
0,0,0,0,0,0,0,0,1), nrow=9,ncol=9,byrow=TRUE)
q<- a[c(TRUE,TRUE,TRUE,TRUE,TRUE,TRUE,TRUE,FALSE,FALSE),c(TRUE,TRUE,TRUE,TRUE,TRUE,TRUE,TRUE,FALSE,FALSE)]
r<- a[c(TRUE,TRUE,TRUE,TRUE,TRUE,TRUE,TRUE,FALSE,FALSE),c(FALSE,FALSE,FALSE,FALSE,FALSE,FALSE,FALSE,TRUE,TRUE)]
ident <- diag(7)
n <- inv(ident - q)
b<- n%*%r
sol<- b[1,2]
print(sol)## [1] 0.02030135There you have it. Smith has a 2% chance of escaping jail this way.
part b)
Since Smith is attempting to double his money each time he cannot hit every integer before 8. Starting at 1, he can go to 0, or 2. From 2 he either can go to 4 or 0. Finally from 4 he can go to 8 or to 0.
We will use the same strategy but with a smaller matrix.
a <- matrix(c(0, 0.4, 0, 0.6, 0,
0, 0, 0.4, 0.6, 0,
0, 0, 0, 0.6, 0.4,
0, 0, 0, 1, 0,
0, 0, 0, 0, 1), nrow= 5, ncol=5, byrow=TRUE)
q<-a[c(TRUE,TRUE,TRUE,FALSE,FALSE), c(TRUE, TRUE, TRUE, FALSE, FALSE)]
r <- a[c(TRUE,TRUE,TRUE,FALSE,FALSE),c(FALSE,FALSE,FALSE,TRUE,TRUE)]
ident <- diag(3)
n<- inv(ident - q)
b<- n%*%r
sol<- b[1,2]
print(sol)## [1] 0.064Using the bold strategy Smith has a 6.4% chance of escaping jail.
part c)
History favors the bold. Smith has a better chance of getting freedom if he bets as bold as possible every time.